Linear evaluations with tic-tac-toe, some data

chrisw · Post by **chrisw** » Fri Sep 04, 2020 11:52 am

Following the discussions on linear evaluations to 'solve' chess, and the unanswered challenge to provide some data for even the simple game of noughts and crosses, or tic-tac-toe as it is known, I did some quick experiments (Python, Keras)

1. Build the entire game tree, evaluating each terminal node, and backing the result up the tree (minimax). This finds 549945 possible board positions, each one with a game result.

2. To make life easier for a linear polynomial to express the result from any position, those 549945 positions were split into groups based on number of men on the board, nine groups.

3. To make life even easier, each group was cut down to remove all symmetrical positions. For example, at ply one, there are 9 positions, but this reduces to 3 once symmetries are taken out. Ply 5 reduces from 15120 positions to 174 and so on.

print("Noughts and crosses")
print("===================")
print("total possible boards from start position 549945")

print("ply 1 total non-mirror boards 3 of 9")
print("ply 2 total non-mirror boards 12 of 72")
print("ply 3 total non-mirror boards 38 of 504")
print("ply 4 total non-mirror boards 108 of 3024")
print("ply 5 total non-mirror boards 174 of 15120")
print("ply 6 total non-mirror boards 204 of 54720")
print("ply 7 total non-mirror boards 154 of 148176")
print("ply 8 total non-mirror boards 59 of 200448")
print("ply 9 total non-mirror boards 15 of 127872")
print()

4. For each ply depth, a network was (relatively exhaustively) trained, with position as input (18 cells, first 9 set to 1 if 'O', second nine set to 1 if 'X', and game result either 0.0, 0.5 or 1.0

5. First results are for a network with no hidden layer, eg a linear polynomial. At nply=1, a network will produce constant results, of course, all ply one positions are draws. A linear polynomial can disentangle the 12 cases at ply 2, and the 15 cases at ply 9, but intermediate plies not.

print("Results, no hidden layer, 2x9 inputs")
print("====================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 6")
print("nply 4 cases 108 errors 22")
print("nply 5 cases 174 errors 98")
print("nply 6 cases 204 errors 78")
print("nply 7 cases 154 errors 85")
print("nply 8 cases 59 errors 36")
print("nply 9 cases 15 errors 0")
print()

6. Next, a hidden layer is added (9 neurons). This improves the results, but is still not able to resolve the all non-linearities.

# 18 x 9 x 1 lots of errors
print("Results, one hidden layer of 9, 2x9 inputs")
print("==========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 12")
print("nply 5 cases 174 errors 8")
print("nply 6 cases 204 errors 5")
print("nply 7 cases 154 errors 10")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

7. Increasing the hidden layer size gives better results, but still some errors. Possibly more runs or tweaking learning rate and so on might reduce errors to zero. I'ld posit that the non-linearities are relatively non-trivial, even for tic-tac-toe, if they need more hidden layer width to resolve.

# 18 x 18 x 1 still a few errors
print("Results, one hidden layer of 18, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 1")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

8. Doubling again the hidden layer size, but still some errors.

# 18 x 36 x 1 still a few errors
print("Results, one hidden layer of 36, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 1")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")

9. Adding another hidden layer (8 neurons) and errors still there.

# still some errors with an added hidden layer
print("Results, two hidden layers of 18 x 8, 2x9 inputs")
print("======================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

10. Create non-linear inputs (NNUE style) and try that ....

Probably won't bother. If tac-tac-toe has non-trivial non-linearities, then the extension to the idea that a linear polynomial operating on (undefined, unknown) non-linear inputs could be created to solve the vastly more complex game of chess doesn't have much going for it.

Ply six example input data with game result.

Code: Select all

O X O X O X . . . , -1
O X O X O . X . . , -1
O X O X O . . X . , -1
O X O X O . . . X , -1
O X O X X O . . . , -1
O X O X . O X . . , -1
O X O X . O . X . , -1
O X O X . O . . X , 0
O X O X X . O . . , 1
O X O X . X O . . , -1
O X O X . . O . X , -1
O X O X X . . O . , 0
O X O X . X . O . , -1
O X O X . . X O . , -1
O X O X . . . O X , 0
O X O X X . . . O , -1
O X O X . . X . O , -1
O X O X . . . X O , -1
O X O O X . X . . , 0
O X O O X . . X . , 1
O X O O X . . . X , -1
O X O . X . O X . , 1
O X O . X . O . X , -1
O X O . X . X O . , 0
O X O O . . X X . , 1
O X O O . . X . X , 0
O X O . O . X X . , -1
O X O . O . X . X , 0
O X O . . O X X . , -1
O X O . . . X O X , 0
O X O . . . X X O , -1
O X X O O X . . . , -1
O X X O O . X . . , -1
O X X O O . . X . , -1
O X X O O . . . X , -1
O X X O X O . . . , -1
O X X O . O X . . , -1
O X X O . O . X . , -1
O X X O . O . . X , -1
O X X O X . . O . , -1
O X X O . X . O . , -1
O X X O . . X O . , -1
O X X O . . . O X , -1
O X X O X . . . O , -1
O X X O . X . . O , -1
O X X O . . X . O , -1
O X X O . . . X O , -1
O X . O X O X . . , 1
O X . O X O . X . , 1
O X . O X O . . X , -1
O X . O X X . O . , -1
O X . O X . X O . , 0
O X . O X . . O X , -1
O X . O X X . . O , -1
O X . O X . X . O , 1
O X . O X . . X O , 1
O X . O O X X . . , -1
O X . O O X . X . , -1
O X . O O X . . X , -1
O X . O . X X O . , 0
O X . O . X . O X , -1
O X . O . X X . O , -1
O X . O . X . X O , -1
O X . O O . X X . , -1
O X . O O . X . X , -1
O X . O . O X X . , -1
O X . O . O X . X , -1
O X . O . . X O X , 0
O X . O . . X X O , -1
O X . O O . . X X , -1
O X . O . O . X X , -1
O X X X O O . . . , -1
O X X . O O X . . , -1
O X X . O O . X . , -1
O X X . O O . . X , -1
O X X . O X O . . , -1
O X X . O . O X . , -1
O X X . O . O . X , -1
O X X X O . . O . , -1
O X X . O X . O . , -1
O X X . O . X O . , -1
O X X . O . . O X , 0
O X . X O O . X . , -1
O X . X O O . . X , 0
O X . . O X O X . , -1
O X . . O X O . X , -1
O X . . O X X O . , -1
O X . . O X . O X , 0
O X . . O O X X . , -1
O X . . O O X . X , -1
O X . . O . X O X , 0
O X . . O O . X X , -1
O X X . X O O . . , -1
O X X . . O O X . , -1
O X X . . O O . X , -1
O X X X . O . O . , -1
O X X . X O . O . , -1
O X X . . O X O . , -1
O X X . . O . O X , -1
O X X X . O . . O , -1
O X X . X O . . O , 1
O X X . . O X . O , -1
O X . X X O . O . , -1
O X . X . O . O X , 0
O X . X X O . . O , -1
O X . X . O X . O , -1
O X . . X O O X . , 1
O X . . X O O . X , -1
O X . . X O X O . , 0
O X . . X O . O X , 0
O X . . X O X . O , -1
O X . . . O X O X , 0
O X X . X . O O . , -1
O X X . . X O O . , -1
O X X . . . O O X , -1
O X X . X . O . O , -1
O X X . . X O . O , -1
O X . . X X O O . , -1
O X . . X . O O X , -1
O X . . X X O . O , -1
O X . . . X O O X , -1
O X X . X . . O O , -1
O X X . . . X O O , -1
O X . . X . X O O , 0
O O X O X X . . . , -1
O O X O X . X . . , 1
O O X O X . . X . , -1
O O X O X . . . X , -1
O O X . X O X . . , 1
O O X . X O . X . , 0
O O X . X O . . X , 0
O O X . X X O . . , -1
O O X . X . O . X , -1
O O X . X X . O . , 1
O O X . X . X O . , 1
O O X . X . . O X , 1
O O X . X . X . O , 1
O O X O . X X . . , 1
O O X O . X . X . , -1
O O X O . X . . X , 1
O O X . O X X . . , -1
O O X . O X . X . , -1
O O X . O X . . X , 1
O O X . . X O . X , 1
O O X . . X X O . , -1
O O X . . X . O X , 1
O O X O . . X . X , 1
O O X . O . X X . , -1
O O X . O . X . X , -1
O O X . . O X X . , 1
O O X . . O X . X , 1
O O X . . . X O X , -1
O O X O . . . X X , -1
O O X . O . . X X , 1
O O X . . O . X X , 0
O . X O X O . X . , -1
O . X O X O . . X , -1
O . X O X X . O . , -1
O . X O X . . O X , -1
O . X O O X . X . , -1
O . X O O X . . X , 1
O . X O . X . O X , 1
O . X O O . . X X , -1
O . X O . O . X X , -1
O . X . O X O . X , 1
O . X . O X X O . , -1
O . X . O X . O X , 1
O . X . O O X . X , -1
O . X . O O . X X , -1
O . X . X O O . X , -1
O . X . X O X O . , 1
O . X . X O . O X , 0
O . X . . O X O X , -1
O . X . X X O O . , -1
O O . O X X . X . , -1
O O . O X X . . X , -1
O O . . X X . O X , -1
O O . . X O . X X , -1
O O . O . X . X X , -1
O O . . O X . X X , -1
X O X O O X . . . , -1
X O X O O . X . . , -1
X O X O O . . X . , -1
X O X O O . . . X , -1
X O X O X O . . . , 1
X O X O . O X . . , -1
X O X O . O . X . , -1
X O X O X . . O . , 1
X O X O . X . O . , -1
X O X O . . . O X , -1
X O . O X O . X . , 0
X O . O X O . . X , 1
X O . O O X . X . , 0
X O . O O X . . X , -1
X O . O . X . O X , -1
X O . X O O X . . , 1
X O . X O O . X . , 0
X O . X O O . . X , -1
X O . . O O X X . , -1
X O . . O O X . X , -1
X O . X X O . O . , 1
X O . X . O X O . , 1
X O . . X O X O . , 1
. O . X O O X X . , 1

towforce · Post by **towforce** » Fri Sep 04, 2020 4:47 pm

chrisw wrote: ↑Fri Sep 04, 2020 11:52 am Following the discussions on linear evaluations to 'solve' chess, and the unanswered challenge to provide some data for even the simple game of noughts and crosses, or tic-tac-toe as it is known, I did some quick experiments (Python, Keras)

1. Build the entire game tree, evaluating each terminal node, and backing the result up the tree (minimax). This finds 549945 possible board positions, each one with a game result.

2. To make life easier for a linear polynomial to express the result from any position, those 549945 positions were split into groups based on number of men on the board, nine groups.

3. To make life even easier, each group was cut down to remove all symmetrical positions. For example, at ply one, there are 9 positions, but this reduces to 3 once symmetries are taken out. Ply 5 reduces from 15120 positions to 174 and so on.

print("Noughts and crosses")
print("===================")
print("total possible boards from start position 549945")

print("ply 1 total non-mirror boards 3 of 9")
print("ply 2 total non-mirror boards 12 of 72")
print("ply 3 total non-mirror boards 38 of 504")
print("ply 4 total non-mirror boards 108 of 3024")
print("ply 5 total non-mirror boards 174 of 15120")
print("ply 6 total non-mirror boards 204 of 54720")
print("ply 7 total non-mirror boards 154 of 148176")
print("ply 8 total non-mirror boards 59 of 200448")
print("ply 9 total non-mirror boards 15 of 127872")
print()

4. For each ply depth, a network was (relatively exhaustively) trained, with position as input (18 cells, first 9 set to 1 if 'O', second nine set to 1 if 'X', and game result either 0.0, 0.5 or 1.0

5. First results are for a network with no hidden layer, eg a linear polynomial. At nply=1, a network will produce constant results, of course, all ply one positions are draws. A linear polynomial can disentangle the 12 cases at ply 2, and the 15 cases at ply 9, but intermediate plies not.

print("Results, no hidden layer, 2x9 inputs")
print("====================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 6")
print("nply 4 cases 108 errors 22")
print("nply 5 cases 174 errors 98")
print("nply 6 cases 204 errors 78")
print("nply 7 cases 154 errors 85")
print("nply 8 cases 59 errors 36")
print("nply 9 cases 15 errors 0")
print()

6. Next, a hidden layer is added (9 neurons). This improves the results, but is still not able to resolve the all non-linearities.

# 18 x 9 x 1 lots of errors
print("Results, one hidden layer of 9, 2x9 inputs")
print("==========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 12")
print("nply 5 cases 174 errors 8")
print("nply 6 cases 204 errors 5")
print("nply 7 cases 154 errors 10")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

7. Increasing the hidden layer size gives better results, but still some errors. Possibly more runs or tweaking learning rate and so on might reduce errors to zero. I'ld posit that the non-linearities are relatively non-trivial, even for tic-tac-toe, if they need more hidden layer width to resolve.

# 18 x 18 x 1 still a few errors
print("Results, one hidden layer of 18, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 1")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

8. Doubling again the hidden layer size, but still some errors.

# 18 x 36 x 1 still a few errors
print("Results, one hidden layer of 36, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 1")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")

9. Adding another hidden layer (8 neurons) and errors still there.

# still some errors with an added hidden layer
print("Results, two hidden layers of 18 x 8, 2x9 inputs")
print("======================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

10. Create non-linear inputs (NNUE style) and try that ....

Probably won't bother. If tac-tac-toe has non-trivial non-linearities, then the extension to the idea that a linear polynomial operating on (undefined, unknown) non-linear inputs could be created to solve the vastly more complex game of chess doesn't have much going for it.

Ply six example input data with game result.
Code: Select all
O X O X O X . . . , -1
O X O X O . X . . , -1
O X O X O . . X . , -1
O X O X O . . . X , -1
O X O X X O . . . , -1
O X O X . O X . . , -1
O X O X . O . X . , -1
O X O X . O . . X , 0
O X O X X . O . . , 1
O X O X . X O . . , -1
O X O X . . O . X , -1
O X O X X . . O . , 0
O X O X . X . O . , -1
O X O X . . X O . , -1
O X O X . . . O X , 0
O X O X X . . . O , -1
O X O X . . X . O , -1
O X O X . . . X O , -1
O X O O X . X . . , 0
O X O O X . . X . , 1
O X O O X . . . X , -1
O X O . X . O X . , 1
O X O . X . O . X , -1
O X O . X . X O . , 0
O X O O . . X X . , 1
O X O O . . X . X , 0
O X O . O . X X . , -1
O X O . O . X . X , 0
O X O . . O X X . , -1
O X O . . . X O X , 0
O X O . . . X X O , -1
O X X O O X . . . , -1
O X X O O . X . . , -1
O X X O O . . X . , -1
O X X O O . . . X , -1
O X X O X O . . . , -1
O X X O . O X . . , -1
O X X O . O . X . , -1
O X X O . O . . X , -1
O X X O X . . O . , -1
O X X O . X . O . , -1
O X X O . . X O . , -1
O X X O . . . O X , -1
O X X O X . . . O , -1
O X X O . X . . O , -1
O X X O . . X . O , -1
O X X O . . . X O , -1
O X . O X O X . . , 1
O X . O X O . X . , 1
O X . O X O . . X , -1
O X . O X X . O . , -1
O X . O X . X O . , 0
O X . O X . . O X , -1
O X . O X X . . O , -1
O X . O X . X . O , 1
O X . O X . . X O , 1
O X . O O X X . . , -1
O X . O O X . X . , -1
O X . O O X . . X , -1
O X . O . X X O . , 0
O X . O . X . O X , -1
O X . O . X X . O , -1
O X . O . X . X O , -1
O X . O O . X X . , -1
O X . O O . X . X , -1
O X . O . O X X . , -1
O X . O . O X . X , -1
O X . O . . X O X , 0
O X . O . . X X O , -1
O X . O O . . X X , -1
O X . O . O . X X , -1
O X X X O O . . . , -1
O X X . O O X . . , -1
O X X . O O . X . , -1
O X X . O O . . X , -1
O X X . O X O . . , -1
O X X . O . O X . , -1
O X X . O . O . X , -1
O X X X O . . O . , -1
O X X . O X . O . , -1
O X X . O . X O . , -1
O X X . O . . O X , 0
O X . X O O . X . , -1
O X . X O O . . X , 0
O X . . O X O X . , -1
O X . . O X O . X , -1
O X . . O X X O . , -1
O X . . O X . O X , 0
O X . . O O X X . , -1
O X . . O O X . X , -1
O X . . O . X O X , 0
O X . . O O . X X , -1
O X X . X O O . . , -1
O X X . . O O X . , -1
O X X . . O O . X , -1
O X X X . O . O . , -1
O X X . X O . O . , -1
O X X . . O X O . , -1
O X X . . O . O X , -1
O X X X . O . . O , -1
O X X . X O . . O , 1
O X X . . O X . O , -1
O X . X X O . O . , -1
O X . X . O . O X , 0
O X . X X O . . O , -1
O X . X . O X . O , -1
O X . . X O O X . , 1
O X . . X O O . X , -1
O X . . X O X O . , 0
O X . . X O . O X , 0
O X . . X O X . O , -1
O X . . . O X O X , 0
O X X . X . O O . , -1
O X X . . X O O . , -1
O X X . . . O O X , -1
O X X . X . O . O , -1
O X X . . X O . O , -1
O X . . X X O O . , -1
O X . . X . O O X , -1
O X . . X X O . O , -1
O X . . . X O O X , -1
O X X . X . . O O , -1
O X X . . . X O O , -1
O X . . X . X O O , 0
O O X O X X . . . , -1
O O X O X . X . . , 1
O O X O X . . X . , -1
O O X O X . . . X , -1
O O X . X O X . . , 1
O O X . X O . X . , 0
O O X . X O . . X , 0
O O X . X X O . . , -1
O O X . X . O . X , -1
O O X . X X . O . , 1
O O X . X . X O . , 1
O O X . X . . O X , 1
O O X . X . X . O , 1
O O X O . X X . . , 1
O O X O . X . X . , -1
O O X O . X . . X , 1
O O X . O X X . . , -1
O O X . O X . X . , -1
O O X . O X . . X , 1
O O X . . X O . X , 1
O O X . . X X O . , -1
O O X . . X . O X , 1
O O X O . . X . X , 1
O O X . O . X X . , -1
O O X . O . X . X , -1
O O X . . O X X . , 1
O O X . . O X . X , 1
O O X . . . X O X , -1
O O X O . . . X X , -1
O O X . O . . X X , 1
O O X . . O . X X , 0
O . X O X O . X . , -1
O . X O X O . . X , -1
O . X O X X . O . , -1
O . X O X . . O X , -1
O . X O O X . X . , -1
O . X O O X . . X , 1
O . X O . X . O X , 1
O . X O O . . X X , -1
O . X O . O . X X , -1
O . X . O X O . X , 1
O . X . O X X O . , -1
O . X . O X . O X , 1
O . X . O O X . X , -1
O . X . O O . X X , -1
O . X . X O O . X , -1
O . X . X O X O . , 1
O . X . X O . O X , 0
O . X . . O X O X , -1
O . X . X X O O . , -1
O O . O X X . X . , -1
O O . O X X . . X , -1
O O . . X X . O X , -1
O O . . X O . X X , -1
O O . O . X . X X , -1
O O . . O X . X X , -1
X O X O O X . . . , -1
X O X O O . X . . , -1
X O X O O . . X . , -1
X O X O O . . . X , -1
X O X O X O . . . , 1
X O X O . O X . . , -1
X O X O . O . X . , -1
X O X O X . . O . , 1
X O X O . X . O . , -1
X O X O . . . O X , -1
X O . O X O . X . , 0
X O . O X O . . X , 1
X O . O O X . X . , 0
X O . O O X . . X , -1
X O . O . X . O X , -1
X O . X O O X . . , 1
X O . X O O . X . , 0
X O . X O O . . X , -1
X O . . O O X X . , -1
X O . . O O X . X , -1
X O . X X O . O . , 1
X O . X . O X O . , 1
X O . . X O X O . , 1
. O . X O O X X . , 1

Congratulations! You have just demonstrated that using NNs to evaluate complex game positions is a bad idea.

Had you fitted a polynomial to all the positions, the resulting EF would always give you the best answer.

Btw, a "linear polynomial" is a polynomial of degree 1. At degree 2 or above, it becomes non linear.

Alayan · Post by **Alayan** » Fri Sep 04, 2020 5:15 pm

A NN isn't a polynomial, but you should actually show a polynomial that "solves" tic-tac-toe if you don't want to keep being laughed at in this forum.

towforce · Post by **towforce** » Fri Sep 04, 2020 5:21 pm

Alayan wrote: ↑Fri Sep 04, 2020 5:15 pm A NN isn't a polynomial, but you should actually show a polynomial that "solves" tic-tac-toe if you don't want to keep being laughed at in this forum.

All in good time. Actually, I don't mind people laughing at me, and I am impressed at how long it takes intelligent people to work this out about me.

chrisw · Post by **chrisw** » Fri Sep 04, 2020 5:36 pm

towforce wrote: ↑Fri Sep 04, 2020 4:47 pm
chrisw wrote: ↑Fri Sep 04, 2020 11:52 am Following the discussions on linear evaluations to 'solve' chess, and the unanswered challenge to provide some data for even the simple game of noughts and crosses, or tic-tac-toe as it is known, I did some quick experiments (Python, Keras)

1. Build the entire game tree, evaluating each terminal node, and backing the result up the tree (minimax). This finds 549945 possible board positions, each one with a game result.

2. To make life easier for a linear polynomial to express the result from any position, those 549945 positions were split into groups based on number of men on the board, nine groups.

3. To make life even easier, each group was cut down to remove all symmetrical positions. For example, at ply one, there are 9 positions, but this reduces to 3 once symmetries are taken out. Ply 5 reduces from 15120 positions to 174 and so on.

print("Noughts and crosses")
print("===================")
print("total possible boards from start position 549945")

print("ply 1 total non-mirror boards 3 of 9")
print("ply 2 total non-mirror boards 12 of 72")
print("ply 3 total non-mirror boards 38 of 504")
print("ply 4 total non-mirror boards 108 of 3024")
print("ply 5 total non-mirror boards 174 of 15120")
print("ply 6 total non-mirror boards 204 of 54720")
print("ply 7 total non-mirror boards 154 of 148176")
print("ply 8 total non-mirror boards 59 of 200448")
print("ply 9 total non-mirror boards 15 of 127872")
print()

4. For each ply depth, a network was (relatively exhaustively) trained, with position as input (18 cells, first 9 set to 1 if 'O', second nine set to 1 if 'X', and game result either 0.0, 0.5 or 1.0

5. First results are for a network with no hidden layer, eg a linear polynomial. At nply=1, a network will produce constant results, of course, all ply one positions are draws. A linear polynomial can disentangle the 12 cases at ply 2, and the 15 cases at ply 9, but intermediate plies not.

print("Results, no hidden layer, 2x9 inputs")
print("====================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 6")
print("nply 4 cases 108 errors 22")
print("nply 5 cases 174 errors 98")
print("nply 6 cases 204 errors 78")
print("nply 7 cases 154 errors 85")
print("nply 8 cases 59 errors 36")
print("nply 9 cases 15 errors 0")
print()

6. Next, a hidden layer is added (9 neurons). This improves the results, but is still not able to resolve the all non-linearities.

# 18 x 9 x 1 lots of errors
print("Results, one hidden layer of 9, 2x9 inputs")
print("==========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 12")
print("nply 5 cases 174 errors 8")
print("nply 6 cases 204 errors 5")
print("nply 7 cases 154 errors 10")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

7. Increasing the hidden layer size gives better results, but still some errors. Possibly more runs or tweaking learning rate and so on might reduce errors to zero. I'ld posit that the non-linearities are relatively non-trivial, even for tic-tac-toe, if they need more hidden layer width to resolve.

# 18 x 18 x 1 still a few errors
print("Results, one hidden layer of 18, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 1")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

8. Doubling again the hidden layer size, but still some errors.

# 18 x 36 x 1 still a few errors
print("Results, one hidden layer of 36, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 1")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")

9. Adding another hidden layer (8 neurons) and errors still there.

# still some errors with an added hidden layer
print("Results, two hidden layers of 18 x 8, 2x9 inputs")
print("======================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

10. Create non-linear inputs (NNUE style) and try that ....

Probably won't bother. If tac-tac-toe has non-trivial non-linearities, then the extension to the idea that a linear polynomial operating on (undefined, unknown) non-linear inputs could be created to solve the vastly more complex game of chess doesn't have much going for it.

Congratulations! You have just demonstrated that using NNs to evaluate complex game positions is a bad idea.

Had you fitted a polynomial to all the positions, the resulting EF would always give you the best answer.

Btw, a "linear polynomial" is a polynomial of degree 1. At degree 2 or above, it becomes non linear.

Well, ignoring the agnorance, the challenge for you a few days ago was to "solve" tic-tac-toe before claiming 8*8 chess was "solvable" by polynomial. Here's the list of all possible unique tic-tac-toe positions as described, just to help you along. We'll await your polynomial expression with interest.

Code: Select all

O . . . . . . . . , 0
. O . . . . . . . , 0
. . . . O . . . . , 0
O X . . . . . . . , -1
O . X . . . . . . , -1
O . . . X . . . . , 0
O . . . . X . . . , -1
O . . . . . . . X , -1
X O . . . . . . . , 0
. O . X . . . . . , -1
. O . . X . . . . , 0
. O . . . . X . . , -1
. O . . . . . X . , 0
X . . . O . . . . , 0
. X . . O . . . . , -1
O X O . . . . . . , 0
O X . O . . . . . , 1
O X . . O . . . . , 1
O X . . . O . . . , 0
O X . . . . O . . , 1
O X . . . . . O . , 0
O X . . . . . . O , 0
O O X . . . . . . , -1
O . X O . . . . . , 1
O . X . O . . . . , 0
O . X . . O . . . , 0
O . X . . . O . . , 1
O . X . . . . O . , 0
O . X . . . . . O , 1
O O . . X . . . . , 0
O . O . X . . . . , 0
O . . . X O . . . , 0
O . . . X . . . O , 0
O O . . . X . . . , -1
O . . O . X . . . , 0
O . . . O X . . . , 1
O . . . . X O . . , 1
O . . . . X . O . , 0
O O . . . . . . X , -1
O . . . O . . . X , 0
O . . . . O . . X , 0
X O . O . . . . . , 0
X O . . O . . . . , 0
X O . . . O . . . , -1
X O . . . . . O . , -1
. O . X O . . . . , 1
. O . X . O . . . , 0
. O . X . . . O . , -1
. O . O X . . . . , 0
. O . . X . . O . , -1
. O . . O . X . . , 0
. O . . . O X . . , -1
. O . . O . . X . , 0
O X O X . . . . . , -1
O X O . X . . . . , 0
O X O . . . X . . , -1
O X O . . . . X . , -1
O X X O . . . . . , -1
O X . O X . . . . , -1
O X . O . X . . . , -1
O X . O . . X . . , -1
O X . O . . . X . , -1
O X . O . . . . X , -1
O X X . O . . . . , -1
O X . X O . . . . , -1
O X . . O X . . . , -1
O X . . O . X . . , -1
O X . . O . . X . , -1
O X . . O . . . X , -1
O X X . . O . . . , -1
O X . X . O . . . , -1
O X . . X O . . . , 0
O X . . . O X . . , -1
O X . . . O . X . , -1
O X . . . O . . X , -1
O X X . . . O . . , -1
O X . . X . O . . , -1
O X . . . X O . . , -1
O X . . . . O X . , -1
O X . . . . O . X , -1
O X X . . . . O . , -1
O X . . X . . O . , -1
O X . . . X . O . , -1
O X . . . . X O . , 0
O X . . . . . O X , 0
O X X . . . . . O , -1
O X . X . . . . O , -1
O X . . X . . . O , 0
O X . . . X . . O , -1
O X . . . . X . O , -1
O X . . . . . X O , -1
O O X . X . . . . , 0
O O X . . X . . . , 1
O O X . . . X . . , -1
O O X . . . . X . , 0
O O X . . . . . X , 1
O . X O X . . . . , -1
O . X O . X . . . , -1
O . X O . . . X . , -1
O . X O . . . . X , -1
O . X . O X . . . , -1
O . X . O . X . . , -1
O . X . O . . X . , -1
O . X . O . . . X , 0
O . X . X O . . . , 0
O . X . . O X . . , -1
O . X . . O . X . , -1
O . X . . O . . X , -1
O . X . X . O . . , -1
O . X . . X O . . , -1
O . X . . . O . X , -1
O . X . X . . O . , -1
O . X . . X . O . , -1
O . X . . . . O X , 0
O . X . X . . . O , -1
O . X . . . X . O , -1
O O . . X X . . . , -1
O O . . X . . X . , -1
O O . . X . . . X , -1
O . O . X . . X . , -1
O . . . X O . X . , 0
O . . . X O . . X , 0
O O . . . X . X . , -1
O O . . . X . . X , -1
O . . O . X . . X , -1
O . . . O X . X . , -1
O . . . O X . . X , -1
O . . . . X . O X , -1
X O X O . . . . . , -1
X O . O X . . . . , 0
X O . O . X . . . , 0
X O . O . . . . X , -1
X O X . O . . . . , -1
X O . X O . . . . , -1
X O . . O X . . . , -1
X O . . O . X . . , -1
X O . . O . . X . , 0
X O . . O . . . X , -1
X O . X . O . . . , -1
X O . . X O . . . , 0
X O . . . O X . . , 1
X O . . . O . X . , 0
X O . . . O . . X , -1
X O X . . . . O . , -1
X O . X . . . O . , -1
X O . . X . . O . , 1
X O . . . X . O . , -1
X O . . . . X O . , -1
X O . . . . . O X , -1
. O . X O X . . . , -1
. O . X O . X . . , -1
. O . X O . . X . , -1
. O . X O . . . X , -1
. O . X X O . . . , -1
. O . X . O X . . , -1
. O . X . O . X . , -1
. O . X X . . O . , 1
. O . X . X . O . , -1
. O . O X . . . X , -1
. O . . O . X X . , 0
. O . . O . X . X , -1
O X O X O . . . . , 1
O X O X . O . . . , 0
O X O X . . O . . , -1
O X O X . . . O . , 0
O X O X . . . . O , 1
O X O O X . . . . , -1
O X O . X . O . . , -1
O X O . X . . O . , 0
O X O O . . X . . , -1
O X O . O . X . . , 0
O X O . . O X . . , 0
O X O . . . X O . , 0
O X O . . . X . O , 1
O X O O . . . X . , -1
O X O . O . . X . , 1
O X O . . . O X . , -1
O X X O O . . . . , 1
O X X O . O . . . , 1
O X X O . . O . . , 1
O X X O . . . O . , 1
O X X O . . . . O , 1
O X . O X O . . . , -1
O X . O X . O . . , 1
O X . O X . . O . , 0
O X . O X . . . O , -1
O X . O O X . . . , 1
O X . O . X O . . , 1
O X . O . X . O . , 0
O X . O . X . . O , 1
O X . O O . X . . , 1
O X . O . O X . . , -1
O X . O . . X O . , 0
O X . O . . X . O , -1
O X . O O . . X . , 1
O X . O . O . X . , -1
O X . O . . O X . , 1
O X . O . . . X O , -1
O X . O O . . . X , 1
O X . O . O . . X , 1
O X . O . . O . X , 1
O X . O . . . O X , 0
O X X . O O . . . , 1
O X X . O . O . . , 1
O X X . O . . O . , 0
O X X . O . . . O , 1
O X . X O O . . . , 0
O X . X O . . . O , 1
O X . . O X O . . , 1
O X . . O X . O . , 0
O X . . O X . . O , 1
O X . . O O X . . , 1
O X . . O . X O . , 0
O X . . O . X . O , 1
O X . . O O . X . , 1
O X . . O . O X . , 1
O X . . O . . X O , 1
O X . . O O . . X , 0
O X . . O . O . X , 1
O X . . O . . O X , 0
O X X . . O O . . , 1
O X X . . O . O . , 1
O X X . . O . . O , -1
O X . X . O . O . , 0
O X . X . O . . O , 1
O X . . X O O . . , -1
O X . . X O . O . , 0
O X . . X O . . O , -1
O X . . . O X O . , 0
O X . . . O X . O , 1
O X . . . O O X . , -1
O X . . . O O . X , 0
O X . . . O . O X , 0
O X X . . . O O . , 1
O X X . . . O . O , 1
O X . . X . O O . , 1
O X . . X . O . O , -1
O X . . . X O O . , 1
O X . . . X O . O , 1
O X . . . . O O X , 0
O X X . . . . O O , 1
O X . . X . . O O , 0
O X . . . . X O O , 0
O O X O X . . . . , -1
O O X . X O . . . , -1
O O X . X . O . . , 0
O O X . X . . O . , -1
O O X . X . . . O , -1
O O X O . X . . . , -1
O O X . O X . . . , -1
O O X . . X O . . , -1
O O X . . X . O . , -1
O O X O . . X . . , -1
O O X . O . X . . , 1
O O X . . O X . . , -1
O O X . . . X O . , -1
O O X . . . X . O , -1
O O X O . . . X . , -1
O O X . O . . X . , -1
O O X . . O . X . , -1
O O X O . . . . X , -1
O O X . O . . . X , -1
O O X . . O . . X , -1
O O X . . . O . X , -1
O O X . . . . O X , -1
O . X O X O . . . , -1
O . X O X . O . . , 1
O . X O X . . O . , -1
O . X O X . . . O , -1
O . X O O X . . . , -1
O . X O . X O . . , 1
O . X O . X . O . , -1
O . X O O . . X . , 1
O . X O . O . X . , 1
O . X O O . . . X , -1
O . X O . O . . X , 1
O . X O . . O . X , 1
O . X O . . . O X , -1
O . X . O X O . . , -1
O . X . O X . O . , -1
O . X . O O X . . , 1
O . X . O . X . O , 1
O . X . O O . X . , 1
O . X . O O . . X , 0
O . X . O . O . X , -1
O . X . O . . O X , -1
O . X . X O O . . , 0
O . X . X O . O . , -1
O . X . . O X O . , -1
O . X . . O O . X , 0
O . X . . O . O X , 0
O . X . X . O O . , 1
O . X . X . O . O , 1
O . X . . X O O . , -1
O O . O X X . . . , 1
O O . . X X O . . , -1
O O . . X X . O . , -1
O O O . X . . X . , 1
O O . . X O . X . , 0
O O . O X . . . X , 1
O O . . X O . . X , 0
O O . . X . . O X , -1
O . . . X O . O X , 0
O O . O . X . X . , 1
O O . . O X . X . , 1
O O . O . X . . X , -1
O O . . O X . . X , -1
O O . . . X . O X , -1
O . . O O X . . X , -1
O . . O . X . O X , -1
O . . . O X . O X , -1
X O X O O . . . . , 1
X O X O . O . . . , -1
X O X O . . . O . , -1
X O . O X O . . . , -1
X O . O O X . . . , 0
X O . O . X . O . , 0
X O . O O . . . X , 1
X O . O . O . . X , -1
X O X . O . . O . , 1
X O . X O O . . . , -1
X O . X O . . O . , 1
X O . . O X . O . , 1
X O . . O O X . . , -1
X O . . O . X O . , 1
X O . . O O . X . , 0
X O . . O O . . X , 1
X O . . O . . O X , 1
X O . X . O . O . , -1
X O . . X O . O . , -1
X O . . . O X O . , -1
. O . X O X . O . , 1
. O . X O O X . . , -1
. O . X O O . X . , -1
. O . X X O . O . , -1
O X O X O X . . . , -1
O X O X O . X . . , -1
O X O X O . . X . , -1
O X O X O . . . X , -1
O X O X X O . . . , -1
O X O X . O X . . , -1
O X O X . O . X . , -1
O X O X . O . . X , 0
O X O X X . O . . , 1
O X O X . X O . . , -1
O X O X . . O . X , -1
O X O X X . . O . , 0
O X O X . X . O . , -1
O X O X . . X O . , -1
O X O X . . . O X , 0
O X O X X . . . O , -1
O X O X . . X . O , -1
O X O X . . . X O , -1
O X O O X . X . . , 0
O X O O X . . X . , 1
O X O O X . . . X , -1
O X O . X . O X . , 1
O X O . X . O . X , -1
O X O . X . X O . , 0
O X O O . . X X . , 1
O X O O . . X . X , 0
O X O . O . X X . , -1
O X O . O . X . X , 0
O X O . . O X X . , -1
O X O . . . X O X , 0
O X O . . . X X O , -1
O X X O O X . . . , -1
O X X O O . X . . , -1
O X X O O . . X . , -1
O X X O O . . . X , -1
O X X O X O . . . , -1
O X X O . O X . . , -1
O X X O . O . X . , -1
O X X O . O . . X , -1
O X X O X . . O . , -1
O X X O . X . O . , -1
O X X O . . X O . , -1
O X X O . . . O X , -1
O X X O X . . . O , -1
O X X O . X . . O , -1
O X X O . . X . O , -1
O X X O . . . X O , -1
O X . O X O X . . , 1
O X . O X O . X . , 1
O X . O X O . . X , -1
O X . O X X . O . , -1
O X . O X . X O . , 0
O X . O X . . O X , -1
O X . O X X . . O , -1
O X . O X . X . O , 1
O X . O X . . X O , 1
O X . O O X X . . , -1
O X . O O X . X . , -1
O X . O O X . . X , -1
O X . O . X X O . , 0
O X . O . X . O X , -1
O X . O . X X . O , -1
O X . O . X . X O , -1
O X . O O . X X . , -1
O X . O O . X . X , -1
O X . O . O X X . , -1
O X . O . O X . X , -1
O X . O . . X O X , 0
O X . O . . X X O , -1
O X . O O . . X X , -1
O X . O . O . X X , -1
O X X X O O . . . , -1
O X X . O O X . . , -1
O X X . O O . X . , -1
O X X . O O . . X , -1
O X X . O X O . . , -1
O X X . O . O X . , -1
O X X . O . O . X , -1
O X X X O . . O . , -1
O X X . O X . O . , -1
O X X . O . X O . , -1
O X X . O . . O X , 0
O X . X O O . X . , -1
O X . X O O . . X , 0
O X . . O X O X . , -1
O X . . O X O . X , -1
O X . . O X X O . , -1
O X . . O X . O X , 0
O X . . O O X X . , -1
O X . . O O X . X , -1
O X . . O . X O X , 0
O X . . O O . X X , -1
O X X . X O O . . , -1
O X X . . O O X . , -1
O X X . . O O . X , -1
O X X X . O . O . , -1
O X X . X O . O . , -1
O X X . . O X O . , -1
O X X . . O . O X , -1
O X X X . O . . O , -1
O X X . X O . . O , 1
O X X . . O X . O , -1
O X . X X O . O . , -1
O X . X . O . O X , 0
O X . X X O . . O , -1
O X . X . O X . O , -1
O X . . X O O X . , 1
O X . . X O O . X , -1
O X . . X O X O . , 0
O X . . X O . O X , 0
O X . . X O X . O , -1
O X . . . O X O X , 0
O X X . X . O O . , -1
O X X . . X O O . , -1
O X X . . . O O X , -1
O X X . X . O . O , -1
O X X . . X O . O , -1
O X . . X X O O . , -1
O X . . X . O O X , -1
O X . . X X O . O , -1
O X . . . X O O X , -1
O X X . X . . O O , -1
O X X . . . X O O , -1
O X . . X . X O O , 0
O O X O X X . . . , -1
O O X O X . X . . , 1
O O X O X . . X . , -1
O O X O X . . . X , -1
O O X . X O X . . , 1
O O X . X O . X . , 0
O O X . X O . . X , 0
O O X . X X O . . , -1
O O X . X . O . X , -1
O O X . X X . O . , 1
O O X . X . X O . , 1
O O X . X . . O X , 1
O O X . X . X . O , 1
O O X O . X X . . , 1
O O X O . X . X . , -1
O O X O . X . . X , 1
O O X . O X X . . , -1
O O X . O X . X . , -1
O O X . O X . . X , 1
O O X . . X O . X , 1
O O X . . X X O . , -1
O O X . . X . O X , 1
O O X O . . X . X , 1
O O X . O . X X . , -1
O O X . O . X . X , -1
O O X . . O X X . , 1
O O X . . O X . X , 1
O O X . . . X O X , -1
O O X O . . . X X , -1
O O X . O . . X X , 1
O O X . . O . X X , 0
O . X O X O . X . , -1
O . X O X O . . X , -1
O . X O X X . O . , -1
O . X O X . . O X , -1
O . X O O X . X . , -1
O . X O O X . . X , 1
O . X O . X . O X , 1
O . X O O . . X X , -1
O . X O . O . X X , -1
O . X . O X O . X , 1
O . X . O X X O . , -1
O . X . O X . O X , 1
O . X . O O X . X , -1
O . X . O O . X X , -1
O . X . X O O . X , -1
O . X . X O X O . , 1
O . X . X O . O X , 0
O . X . . O X O X , -1
O . X . X X O O . , -1
O O . O X X . X . , -1
O O . O X X . . X , -1
O O . . X X . O X , -1
O O . . X O . X X , -1
O O . O . X . X X , -1
O O . . O X . X X , -1
X O X O O X . . . , -1
X O X O O . X . . , -1
X O X O O . . X . , -1
X O X O O . . . X , -1
X O X O X O . . . , 1
X O X O . O X . . , -1
X O X O . O . X . , -1
X O X O X . . O . , 1
X O X O . X . O . , -1
X O X O . . . O X , -1
X O . O X O . X . , 0
X O . O X O . . X , 1
X O . O O X . X . , 0
X O . O O X . . X , -1
X O . O . X . O X , -1
X O . X O O X . . , 1
X O . X O O . X . , 0
X O . X O O . . X , -1
X O . . O O X X . , -1
X O . . O O X . X , -1
X O . X X O . O . , 1
X O . X . O X O . , 1
X O . . X O X O . , 1
. O . X O O X X . , 1
O X O X O X O . . , 1
O X O X O X . O . , 1
O X O X O O X . . , 0
O X O X O . X O . , 0
O X O X O . X . O , 1
O X O X O O . X . , 1
O X O X O . . X O , 1
O X O X O O . . X , 0
O X O X O . O . X , 1
O X O X O . . O X , 0
O X O X X O O . . , -1
O X O X X O . O . , 0
O X O X X O . . O , 1
O X O X . O X O . , 0
O X O X . O X . O , 1
O X O X . O O X . , -1
O X O X . O . X O , 1
O X O X . O O . X , 0
O X O X . O . O X , 0
O X O X X . O . O , -1
O X O X . X O . O , -1
O X O X X . . O O , -1
O X O X . . X O O , 1
O X O O X O X . . , -1
O X O O X . X O . , 0
O X O O X . X . O , -1
O X O O X . O . X , 1
O X O O X . . O X , 0
O X O . X . O O X , 0
O X O O O . X X . , -1
O X O O . O X X . , -1
O X O O . . X X O , -1
O X O O O . X . X , -1
O X O O . O X . X , -1
O X O O . . X O X , 0
O X O . O O X X . , -1
O X O . O . X X O , 1
O X O . O . X O X , 0
O X O . . O X X O , 1
O X X O O X O . . , 1
O X X O O X . O . , -1
O X X O O X . . O , 1
O X X O O O X . . , 1
O X X O O . X O . , 1
O X X O O . X . O , 1
O X X O O O . X . , 1
O X X O O . O X . , 1
O X X O O . . X O , 1
O X X O O O . . X , 1
O X X O O . O . X , 1
O X X O O . . O X , -1
O X X O X O O . . , 1
O X X O X O . O . , -1
O X X O X O . . O , -1
O X X O . O X O . , -1
O X X O . O X . O , -1
O X X O . O O X . , 1
O X X O . O . X O , -1
O X X O . O O . X , 1
O X X O . O . O X , 1
O X X O X . O O . , 1
O X X O X . . O O , -1
O X X O . X O O . , 1
O X X O . X . O O , 1
O X X O . . X O O , -1
O X X O . . O O X , 1
O X X O X . O . O , 1
O X X O . X O . O , 1
O X . O X O X O . , -1
O X . O X O X . O , -1
O X . O X O O . X , 1
O X . O X O . O X , 0
O X . O X X O O . , 1
O X . O X X . O O , 0
O X . O X . X O O , -1
O X . O X . O O X , 1
O X . O X X O . O , 1
O X . O O X X O . , 0
O X . O O X X . O , 1
O X . O O X O X . , 1
O X . O O X . X O , 1
O X . O O X O . X , 1
O X . O O X . O X , -1
O X . O . X X O O , 0
O X . O . X O O X , 1
O X . O O O X X . , 1
O X . O O . X X O , 1
O X . O O O X . X , 1
O X . O O . X O X , 0
O X . O . O X O X , 0
O X . O O O . X X , 1
O X X X O O . O . , 0
O X X X O O . . O , 1
O X X . O O X O . , 1
O X X . O O X . O , 1
O X X . O O O X . , 1
O X X . O O O . X , 0
O X X . O O . O X , 0
O X X . O X O O . , -1
O X X . O X O . O , 1
O X X . O . O O X , -1
O X X X O . . O O , 1
O X X . O . X O O , 1
O X . X O O . O X , 0
O X . . O X O O X , -1
O X . . O O X O X , 0
O X X . X O O O . , 1
O X X . X O O . O , -1
O X X . . O O O X , 0
O X X X . O . O O , 1
O X X . X O . O O , -1
O X X . . O X O O , -1
O X . X X O . O O , 1
O X . . X O O O X , 0
O X . . X O X O O , -1
O X X . X . O O O , 1
O O X O X X O . . , 1
O O X O X X . O . , -1
O O X O X O . X . , -1
O O X O X O . . X , -1
O O X O X . O . X , 1
O O X O X . . O X , -1
O O X . X O O . X , 0
O O X . X O . O X , -1
O O X . X X O O . , -1
O O X . X . O O X , -1
O O X O O X X . . , -1
O O X O . X X O . , -1
O O X O O X . X . , -1
O O X . O X X O . , 1
O O X O O . X . X , -1
O O X O . O X . X , -1
O O X . O O X X . , -1
O O X . O O X . X , -1
O O X . O . X O X , 1
O O X . . O X O X , -1
O O X O O . . X X , -1
O O X O . O . X X , -1
O O X . O O . X X , -1
O . X O X O O . X , 1
O . X O X O . O X , -1
O . X O X X O O . , 1
O . X O O O . X X , 1
O . X . O O X O X , 1
O O . O X X . O X , -1
O O . O O X . X X , -1
X O X O O X . O . , 1
X O X O O O X . . , 1
X O X O O . X O . , 1
X O X O O O . X . , 1
X O X O O . . O X , 1
X O X O X O . O . , -1
X O X O . O X O . , -1
X O . O O X . O X , 1
O X O X O X X O . , -1
O X O X O O X X . , -1
O X O X O O X . X , 0
O X O X O . X O X , 0
O X O X O O . X X , -1
O X O X X O O X . , 1
O X O X X O O . X , 0
O X O X X O X O . , -1
O X O X X O . O X , 0
O X O X . O X O X , 0
O X O X . O O X X , -1
O X O X X X O . O , 1
O X O X X . O X O , 1
O X O X . X O X O , -1
O X O X X . X O O , -1
O X O O X O X X . , 1
O X O O X O X . X , 0
O X O O X . X O X , 0
O X O O X . X X O , 1
O X O O O . X X X , 1
O X O O . O X X X , 1
O X X O O X X O . , -1
O X X O O X . O X , 1
O X X O O . X O X , -1
O X X O X O X O . , 1
O X X O X O . O X , -1
O X X O X O X . O , 1
O X X O X O . X O , 1
O X X O . O X O X , -1
O X X O . O X X O , -1
O X X O X X . O O , -1
O X X O X . X O O , 1
O X X O . X X O O , -1
O X . O X O X O X , 0
O X . O X X X O O , 0
O X . O O X X O X , 0
O X X X O O X O . , -1
O X X X O O . O X , 0
O X X . O O X O X , -1
O X X . O O O X X , -1
O X X . O X O O X , 1
O X X . X O O O X , -1
O X X X X O . O O , -1
O X X X . O X O O , -1
O X X . X O X O O , 1
O O X O X X X O . , 1
O O X O X X . O X , 1
O O X O X O . X X , -1
O O X O X O X . X , 1
O O X . X O X O X , 1
O O X . X X O O X , 1
O O X O O X X X . , -1
O O X O O X X . X , 1
O O X O . X X O X , 1
O O X O O X . X X , 1
O O X . O O X X X , 1
X O X O X O X O . , 1
X O X O X O . O X , 1
X O X O . O X O X , -1
O X O X O X X O O , 1
O X O X O O X X O , 1
O X O X O O X O X , 0
O X O X X O O O X , 0
O X O X X O X O O , 1
O X O X O X O X O , 1
O X O O X O X O X , 0
O X X O O X X O O , 1
O X X O O O X O X , 1
O X X O X O O O X , 1
O X X O O O X X O , 1
O X X O X X O O O , 1
O X X X O O X O O , 1
O X X O O O O X X , 1
X O X O O O X O X , 1

towforce · Post by **towforce** » Fri Sep 04, 2020 5:46 pm

chrisw wrote: ↑Fri Sep 04, 2020 5:36 pmWell, ignoring the agnorance, the challenge for you a few days ago was to "solve" tic-tac-toe before claiming 8*8 chess was "solvable" by polynomial. Here's the list of all possible unique tic-tac-toe positions as described, just to help you along. We'll await your polynomial expression with interest.

Code: Select all

O . . . . . . . . , 0
. O . . . . . . . , 0
. . . . O . . . . , 0
O X . . . . . . . , -1
O . X . . . . . . , -1
O . . . X . . . . , 0
O . . . . X . . . , -1
O . . . . . . . X , -1
X O . . . . . . . , 0
. O . X . . . . . , -1
. O . . X . . . . , 0
. O . . . . X . . , -1
. O . . . . . X . , 0
X . . . O . . . . , 0
. X . . O . . . . , -1
O X O . . . . . . , 0
O X . O . . . . . , 1
O X . . O . . . . , 1
O X . . . O . . . , 0
O X . . . . O . . , 1
O X . . . . . O . , 0
O X . . . . . . O , 0
O O X . . . . . . , -1
O . X O . . . . . , 1
O . X . O . . . . , 0
O . X . . O . . . , 0
O . X . . . O . . , 1
O . X . . . . O . , 0
O . X . . . . . O , 1
O O . . X . . . . , 0
O . O . X . . . . , 0
O . . . X O . . . , 0
O . . . X . . . O , 0
O O . . . X . . . , -1
O . . O . X . . . , 0
O . . . O X . . . , 1
O . . . . X O . . , 1
O . . . . X . O . , 0
O O . . . . . . X , -1
O . . . O . . . X , 0
O . . . . O . . X , 0
X O . O . . . . . , 0
X O . . O . . . . , 0
X O . . . O . . . , -1
X O . . . . . O . , -1
. O . X O . . . . , 1
. O . X . O . . . , 0
. O . X . . . O . , -1
. O . O X . . . . , 0
. O . . X . . O . , -1
. O . . O . X . . , 0
. O . . . O X . . , -1
. O . . O . . X . , 0
O X O X . . . . . , -1
O X O . X . . . . , 0
O X O . . . X . . , -1
O X O . . . . X . , -1
O X X O . . . . . , -1
O X . O X . . . . , -1
O X . O . X . . . , -1
O X . O . . X . . , -1
O X . O . . . X . , -1
O X . O . . . . X , -1
O X X . O . . . . , -1
O X . X O . . . . , -1
O X . . O X . . . , -1
O X . . O . X . . , -1
O X . . O . . X . , -1
O X . . O . . . X , -1
O X X . . O . . . , -1
O X . X . O . . . , -1
O X . . X O . . . , 0
O X . . . O X . . , -1
O X . . . O . X . , -1
O X . . . O . . X , -1
O X X . . . O . . , -1
O X . . X . O . . , -1
O X . . . X O . . , -1
O X . . . . O X . , -1
O X . . . . O . X , -1
O X X . . . . O . , -1
O X . . X . . O . , -1
O X . . . X . O . , -1
O X . . . . X O . , 0
O X . . . . . O X , 0
O X X . . . . . O , -1
O X . X . . . . O , -1
O X . . X . . . O , 0
O X . . . X . . O , -1
O X . . . . X . O , -1
O X . . . . . X O , -1
O O X . X . . . . , 0
O O X . . X . . . , 1
O O X . . . X . . , -1
O O X . . . . X . , 0
O O X . . . . . X , 1
O . X O X . . . . , -1
O . X O . X . . . , -1
O . X O . . . X . , -1
O . X O . . . . X , -1
O . X . O X . . . , -1
O . X . O . X . . , -1
O . X . O . . X . , -1
O . X . O . . . X , 0
O . X . X O . . . , 0
O . X . . O X . . , -1
O . X . . O . X . , -1
O . X . . O . . X , -1
O . X . X . O . . , -1
O . X . . X O . . , -1
O . X . . . O . X , -1
O . X . X . . O . , -1
O . X . . X . O . , -1
O . X . . . . O X , 0
O . X . X . . . O , -1
O . X . . . X . O , -1
O O . . X X . . . , -1
O O . . X . . X . , -1
O O . . X . . . X , -1
O . O . X . . X . , -1
O . . . X O . X . , 0
O . . . X O . . X , 0
O O . . . X . X . , -1
O O . . . X . . X , -1
O . . O . X . . X , -1
O . . . O X . X . , -1
O . . . O X . . X , -1
O . . . . X . O X , -1
X O X O . . . . . , -1
X O . O X . . . . , 0
X O . O . X . . . , 0
X O . O . . . . X , -1
X O X . O . . . . , -1
X O . X O . . . . , -1
X O . . O X . . . , -1
X O . . O . X . . , -1
X O . . O . . X . , 0
X O . . O . . . X , -1
X O . X . O . . . , -1
X O . . X O . . . , 0
X O . . . O X . . , 1
X O . . . O . X . , 0
X O . . . O . . X , -1
X O X . . . . O . , -1
X O . X . . . O . , -1
X O . . X . . O . , 1
X O . . . X . O . , -1
X O . . . . X O . , -1
X O . . . . . O X , -1
. O . X O X . . . , -1
. O . X O . X . . , -1
. O . X O . . X . , -1
. O . X O . . . X , -1
. O . X X O . . . , -1
. O . X . O X . . , -1
. O . X . O . X . , -1
. O . X X . . O . , 1
. O . X . X . O . , -1
. O . O X . . . X , -1
. O . . O . X X . , 0
. O . . O . X . X , -1
O X O X O . . . . , 1
O X O X . O . . . , 0
O X O X . . O . . , -1
O X O X . . . O . , 0
O X O X . . . . O , 1
O X O O X . . . . , -1
O X O . X . O . . , -1
O X O . X . . O . , 0
O X O O . . X . . , -1
O X O . O . X . . , 0
O X O . . O X . . , 0
O X O . . . X O . , 0
O X O . . . X . O , 1
O X O O . . . X . , -1
O X O . O . . X . , 1
O X O . . . O X . , -1
O X X O O . . . . , 1
O X X O . O . . . , 1
O X X O . . O . . , 1
O X X O . . . O . , 1
O X X O . . . . O , 1
O X . O X O . . . , -1
O X . O X . O . . , 1
O X . O X . . O . , 0
O X . O X . . . O , -1
O X . O O X . . . , 1
O X . O . X O . . , 1
O X . O . X . O . , 0
O X . O . X . . O , 1
O X . O O . X . . , 1
O X . O . O X . . , -1
O X . O . . X O . , 0
O X . O . . X . O , -1
O X . O O . . X . , 1
O X . O . O . X . , -1
O X . O . . O X . , 1
O X . O . . . X O , -1
O X . O O . . . X , 1
O X . O . O . . X , 1
O X . O . . O . X , 1
O X . O . . . O X , 0
O X X . O O . . . , 1
O X X . O . O . . , 1
O X X . O . . O . , 0
O X X . O . . . O , 1
O X . X O O . . . , 0
O X . X O . . . O , 1
O X . . O X O . . , 1
O X . . O X . O . , 0
O X . . O X . . O , 1
O X . . O O X . . , 1
O X . . O . X O . , 0
O X . . O . X . O , 1
O X . . O O . X . , 1
O X . . O . O X . , 1
O X . . O . . X O , 1
O X . . O O . . X , 0
O X . . O . O . X , 1
O X . . O . . O X , 0
O X X . . O O . . , 1
O X X . . O . O . , 1
O X X . . O . . O , -1
O X . X . O . O . , 0
O X . X . O . . O , 1
O X . . X O O . . , -1
O X . . X O . O . , 0
O X . . X O . . O , -1
O X . . . O X O . , 0
O X . . . O X . O , 1
O X . . . O O X . , -1
O X . . . O O . X , 0
O X . . . O . O X , 0
O X X . . . O O . , 1
O X X . . . O . O , 1
O X . . X . O O . , 1
O X . . X . O . O , -1
O X . . . X O O . , 1
O X . . . X O . O , 1
O X . . . . O O X , 0
O X X . . . . O O , 1
O X . . X . . O O , 0
O X . . . . X O O , 0
O O X O X . . . . , -1
O O X . X O . . . , -1
O O X . X . O . . , 0
O O X . X . . O . , -1
O O X . X . . . O , -1
O O X O . X . . . , -1
O O X . O X . . . , -1
O O X . . X O . . , -1
O O X . . X . O . , -1
O O X O . . X . . , -1
O O X . O . X . . , 1
O O X . . O X . . , -1
O O X . . . X O . , -1
O O X . . . X . O , -1
O O X O . . . X . , -1
O O X . O . . X . , -1
O O X . . O . X . , -1
O O X O . . . . X , -1
O O X . O . . . X , -1
O O X . . O . . X , -1
O O X . . . O . X , -1
O O X . . . . O X , -1
O . X O X O . . . , -1
O . X O X . O . . , 1
O . X O X . . O . , -1
O . X O X . . . O , -1
O . X O O X . . . , -1
O . X O . X O . . , 1
O . X O . X . O . , -1
O . X O O . . X . , 1
O . X O . O . X . , 1
O . X O O . . . X , -1
O . X O . O . . X , 1
O . X O . . O . X , 1
O . X O . . . O X , -1
O . X . O X O . . , -1
O . X . O X . O . , -1
O . X . O O X . . , 1
O . X . O . X . O , 1
O . X . O O . X . , 1
O . X . O O . . X , 0
O . X . O . O . X , -1
O . X . O . . O X , -1
O . X . X O O . . , 0
O . X . X O . O . , -1
O . X . . O X O . , -1
O . X . . O O . X , 0
O . X . . O . O X , 0
O . X . X . O O . , 1
O . X . X . O . O , 1
O . X . . X O O . , -1
O O . O X X . . . , 1
O O . . X X O . . , -1
O O . . X X . O . , -1
O O O . X . . X . , 1
O O . . X O . X . , 0
O O . O X . . . X , 1
O O . . X O . . X , 0
O O . . X . . O X , -1
O . . . X O . O X , 0
O O . O . X . X . , 1
O O . . O X . X . , 1
O O . O . X . . X , -1
O O . . O X . . X , -1
O O . . . X . O X , -1
O . . O O X . . X , -1
O . . O . X . O X , -1
O . . . O X . O X , -1
X O X O O . . . . , 1
X O X O . O . . . , -1
X O X O . . . O . , -1
X O . O X O . . . , -1
X O . O O X . . . , 0
X O . O . X . O . , 0
X O . O O . . . X , 1
X O . O . O . . X , -1
X O X . O . . O . , 1
X O . X O O . . . , -1
X O . X O . . O . , 1
X O . . O X . O . , 1
X O . . O O X . . , -1
X O . . O . X O . , 1
X O . . O O . X . , 0
X O . . O O . . X , 1
X O . . O . . O X , 1
X O . X . O . O . , -1
X O . . X O . O . , -1
X O . . . O X O . , -1
. O . X O X . O . , 1
. O . X O O X . . , -1
. O . X O O . X . , -1
. O . X X O . O . , -1
O X O X O X . . . , -1
O X O X O . X . . , -1
O X O X O . . X . , -1
O X O X O . . . X , -1
O X O X X O . . . , -1
O X O X . O X . . , -1
O X O X . O . X . , -1
O X O X . O . . X , 0
O X O X X . O . . , 1
O X O X . X O . . , -1
O X O X . . O . X , -1
O X O X X . . O . , 0
O X O X . X . O . , -1
O X O X . . X O . , -1
O X O X . . . O X , 0
O X O X X . . . O , -1
O X O X . . X . O , -1
O X O X . . . X O , -1
O X O O X . X . . , 0
O X O O X . . X . , 1
O X O O X . . . X , -1
O X O . X . O X . , 1
O X O . X . O . X , -1
O X O . X . X O . , 0
O X O O . . X X . , 1
O X O O . . X . X , 0
O X O . O . X X . , -1
O X O . O . X . X , 0
O X O . . O X X . , -1
O X O . . . X O X , 0
O X O . . . X X O , -1
O X X O O X . . . , -1
O X X O O . X . . , -1
O X X O O . . X . , -1
O X X O O . . . X , -1
O X X O X O . . . , -1
O X X O . O X . . , -1
O X X O . O . X . , -1
O X X O . O . . X , -1
O X X O X . . O . , -1
O X X O . X . O . , -1
O X X O . . X O . , -1
O X X O . . . O X , -1
O X X O X . . . O , -1
O X X O . X . . O , -1
O X X O . . X . O , -1
O X X O . . . X O , -1
O X . O X O X . . , 1
O X . O X O . X . , 1
O X . O X O . . X , -1
O X . O X X . O . , -1
O X . O X . X O . , 0
O X . O X . . O X , -1
O X . O X X . . O , -1
O X . O X . X . O , 1
O X . O X . . X O , 1
O X . O O X X . . , -1
O X . O O X . X . , -1
O X . O O X . . X , -1
O X . O . X X O . , 0
O X . O . X . O X , -1
O X . O . X X . O , -1
O X . O . X . X O , -1
O X . O O . X X . , -1
O X . O O . X . X , -1
O X . O . O X X . , -1
O X . O . O X . X , -1
O X . O . . X O X , 0
O X . O . . X X O , -1
O X . O O . . X X , -1
O X . O . O . X X , -1
O X X X O O . . . , -1
O X X . O O X . . , -1
O X X . O O . X . , -1
O X X . O O . . X , -1
O X X . O X O . . , -1
O X X . O . O X . , -1
O X X . O . O . X , -1
O X X X O . . O . , -1
O X X . O X . O . , -1
O X X . O . X O . , -1
O X X . O . . O X , 0
O X . X O O . X . , -1
O X . X O O . . X , 0
O X . . O X O X . , -1
O X . . O X O . X , -1
O X . . O X X O . , -1
O X . . O X . O X , 0
O X . . O O X X . , -1
O X . . O O X . X , -1
O X . . O . X O X , 0
O X . . O O . X X , -1
O X X . X O O . . , -1
O X X . . O O X . , -1
O X X . . O O . X , -1
O X X X . O . O . , -1
O X X . X O . O . , -1
O X X . . O X O . , -1
O X X . . O . O X , -1
O X X X . O . . O , -1
O X X . X O . . O , 1
O X X . . O X . O , -1
O X . X X O . O . , -1
O X . X . O . O X , 0
O X . X X O . . O , -1
O X . X . O X . O , -1
O X . . X O O X . , 1
O X . . X O O . X , -1
O X . . X O X O . , 0
O X . . X O . O X , 0
O X . . X O X . O , -1
O X . . . O X O X , 0
O X X . X . O O . , -1
O X X . . X O O . , -1
O X X . . . O O X , -1
O X X . X . O . O , -1
O X X . . X O . O , -1
O X . . X X O O . , -1
O X . . X . O O X , -1
O X . . X X O . O , -1
O X . . . X O O X , -1
O X X . X . . O O , -1
O X X . . . X O O , -1
O X . . X . X O O , 0
O O X O X X . . . , -1
O O X O X . X . . , 1
O O X O X . . X . , -1
O O X O X . . . X , -1
O O X . X O X . . , 1
O O X . X O . X . , 0
O O X . X O . . X , 0
O O X . X X O . . , -1
O O X . X . O . X , -1
O O X . X X . O . , 1
O O X . X . X O . , 1
O O X . X . . O X , 1
O O X . X . X . O , 1
O O X O . X X . . , 1
O O X O . X . X . , -1
O O X O . X . . X , 1
O O X . O X X . . , -1
O O X . O X . X . , -1
O O X . O X . . X , 1
O O X . . X O . X , 1
O O X . . X X O . , -1
O O X . . X . O X , 1
O O X O . . X . X , 1
O O X . O . X X . , -1
O O X . O . X . X , -1
O O X . . O X X . , 1
O O X . . O X . X , 1
O O X . . . X O X , -1
O O X O . . . X X , -1
O O X . O . . X X , 1
O O X . . O . X X , 0
O . X O X O . X . , -1
O . X O X O . . X , -1
O . X O X X . O . , -1
O . X O X . . O X , -1
O . X O O X . X . , -1
O . X O O X . . X , 1
O . X O . X . O X , 1
O . X O O . . X X , -1
O . X O . O . X X , -1
O . X . O X O . X , 1
O . X . O X X O . , -1
O . X . O X . O X , 1
O . X . O O X . X , -1
O . X . O O . X X , -1
O . X . X O O . X , -1
O . X . X O X O . , 1
O . X . X O . O X , 0
O . X . . O X O X , -1
O . X . X X O O . , -1
O O . O X X . X . , -1
O O . O X X . . X , -1
O O . . X X . O X , -1
O O . . X O . X X , -1
O O . O . X . X X , -1
O O . . O X . X X , -1
X O X O O X . . . , -1
X O X O O . X . . , -1
X O X O O . . X . , -1
X O X O O . . . X , -1
X O X O X O . . . , 1
X O X O . O X . . , -1
X O X O . O . X . , -1
X O X O X . . O . , 1
X O X O . X . O . , -1
X O X O . . . O X , -1
X O . O X O . X . , 0
X O . O X O . . X , 1
X O . O O X . X . , 0
X O . O O X . . X , -1
X O . O . X . O X , -1
X O . X O O X . . , 1
X O . X O O . X . , 0
X O . X O O . . X , -1
X O . . O O X X . , -1
X O . . O O X . X , -1
X O . X X O . O . , 1
X O . X . O X O . , 1
X O . . X O X O . , 1
. O . X O O X X . , 1
O X O X O X O . . , 1
O X O X O X . O . , 1
O X O X O O X . . , 0
O X O X O . X O . , 0
O X O X O . X . O , 1
O X O X O O . X . , 1
O X O X O . . X O , 1
O X O X O O . . X , 0
O X O X O . O . X , 1
O X O X O . . O X , 0
O X O X X O O . . , -1
O X O X X O . O . , 0
O X O X X O . . O , 1
O X O X . O X O . , 0
O X O X . O X . O , 1
O X O X . O O X . , -1
O X O X . O . X O , 1
O X O X . O O . X , 0
O X O X . O . O X , 0
O X O X X . O . O , -1
O X O X . X O . O , -1
O X O X X . . O O , -1
O X O X . . X O O , 1
O X O O X O X . . , -1
O X O O X . X O . , 0
O X O O X . X . O , -1
O X O O X . O . X , 1
O X O O X . . O X , 0
O X O . X . O O X , 0
O X O O O . X X . , -1
O X O O . O X X . , -1
O X O O . . X X O , -1
O X O O O . X . X , -1
O X O O . O X . X , -1
O X O O . . X O X , 0
O X O . O O X X . , -1
O X O . O . X X O , 1
O X O . O . X O X , 0
O X O . . O X X O , 1
O X X O O X O . . , 1
O X X O O X . O . , -1
O X X O O X . . O , 1
O X X O O O X . . , 1
O X X O O . X O . , 1
O X X O O . X . O , 1
O X X O O O . X . , 1
O X X O O . O X . , 1
O X X O O . . X O , 1
O X X O O O . . X , 1
O X X O O . O . X , 1
O X X O O . . O X , -1
O X X O X O O . . , 1
O X X O X O . O . , -1
O X X O X O . . O , -1
O X X O . O X O . , -1
O X X O . O X . O , -1
O X X O . O O X . , 1
O X X O . O . X O , -1
O X X O . O O . X , 1
O X X O . O . O X , 1
O X X O X . O O . , 1
O X X O X . . O O , -1
O X X O . X O O . , 1
O X X O . X . O O , 1
O X X O . . X O O , -1
O X X O . . O O X , 1
O X X O X . O . O , 1
O X X O . X O . O , 1
O X . O X O X O . , -1
O X . O X O X . O , -1
O X . O X O O . X , 1
O X . O X O . O X , 0
O X . O X X O O . , 1
O X . O X X . O O , 0
O X . O X . X O O , -1
O X . O X . O O X , 1
O X . O X X O . O , 1
O X . O O X X O . , 0
O X . O O X X . O , 1
O X . O O X O X . , 1
O X . O O X . X O , 1
O X . O O X O . X , 1
O X . O O X . O X , -1
O X . O . X X O O , 0
O X . O . X O O X , 1
O X . O O O X X . , 1
O X . O O . X X O , 1
O X . O O O X . X , 1
O X . O O . X O X , 0
O X . O . O X O X , 0
O X . O O O . X X , 1
O X X X O O . O . , 0
O X X X O O . . O , 1
O X X . O O X O . , 1
O X X . O O X . O , 1
O X X . O O O X . , 1
O X X . O O O . X , 0
O X X . O O . O X , 0
O X X . O X O O . , -1
O X X . O X O . O , 1
O X X . O . O O X , -1
O X X X O . . O O , 1
O X X . O . X O O , 1
O X . X O O . O X , 0
O X . . O X O O X , -1
O X . . O O X O X , 0
O X X . X O O O . , 1
O X X . X O O . O , -1
O X X . . O O O X , 0
O X X X . O . O O , 1
O X X . X O . O O , -1
O X X . . O X O O , -1
O X . X X O . O O , 1
O X . . X O O O X , 0
O X . . X O X O O , -1
O X X . X . O O O , 1
O O X O X X O . . , 1
O O X O X X . O . , -1
O O X O X O . X . , -1
O O X O X O . . X , -1
O O X O X . O . X , 1
O O X O X . . O X , -1
O O X . X O O . X , 0
O O X . X O . O X , -1
O O X . X X O O . , -1
O O X . X . O O X , -1
O O X O O X X . . , -1
O O X O . X X O . , -1
O O X O O X . X . , -1
O O X . O X X O . , 1
O O X O O . X . X , -1
O O X O . O X . X , -1
O O X . O O X X . , -1
O O X . O O X . X , -1
O O X . O . X O X , 1
O O X . . O X O X , -1
O O X O O . . X X , -1
O O X O . O . X X , -1
O O X . O O . X X , -1
O . X O X O O . X , 1
O . X O X O . O X , -1
O . X O X X O O . , 1
O . X O O O . X X , 1
O . X . O O X O X , 1
O O . O X X . O X , -1
O O . O O X . X X , -1
X O X O O X . O . , 1
X O X O O O X . . , 1
X O X O O . X O . , 1
X O X O O O . X . , 1
X O X O O . . O X , 1
X O X O X O . O . , -1
X O X O . O X O . , -1
X O . O O X . O X , 1
O X O X O X X O . , -1
O X O X O O X X . , -1
O X O X O O X . X , 0
O X O X O . X O X , 0
O X O X O O . X X , -1
O X O X X O O X . , 1
O X O X X O O . X , 0
O X O X X O X O . , -1
O X O X X O . O X , 0
O X O X . O X O X , 0
O X O X . O O X X , -1
O X O X X X O . O , 1
O X O X X . O X O , 1
O X O X . X O X O , -1
O X O X X . X O O , -1
O X O O X O X X . , 1
O X O O X O X . X , 0
O X O O X . X O X , 0
O X O O X . X X O , 1
O X O O O . X X X , 1
O X O O . O X X X , 1
O X X O O X X O . , -1
O X X O O X . O X , 1
O X X O O . X O X , -1
O X X O X O X O . , 1
O X X O X O . O X , -1
O X X O X O X . O , 1
O X X O X O . X O , 1
O X X O . O X O X , -1
O X X O . O X X O , -1
O X X O X X . O O , -1
O X X O X . X O O , 1
O X X O . X X O O , -1
O X . O X O X O X , 0
O X . O X X X O O , 0
O X . O O X X O X , 0
O X X X O O X O . , -1
O X X X O O . O X , 0
O X X . O O X O X , -1
O X X . O O O X X , -1
O X X . O X O O X , 1
O X X . X O O O X , -1
O X X X X O . O O , -1
O X X X . O X O O , -1
O X X . X O X O O , 1
O O X O X X X O . , 1
O O X O X X . O X , 1
O O X O X O . X X , -1
O O X O X O X . X , 1
O O X . X O X O X , 1
O O X . X X O O X , 1
O O X O O X X X . , -1
O O X O O X X . X , 1
O O X O . X X O X , 1
O O X O O X . X X , 1
O O X . O O X X X , 1
X O X O X O X O . , 1
X O X O X O . O X , 1
X O X O . O X O X , -1
O X O X O X X O O , 1
O X O X O O X X O , 1
O X O X O O X O X , 0
O X O X X O O O X , 0
O X O X X O X O O , 1
O X O X O X O X O , 1
O X O O X O X O X , 0
O X X O O X X O O , 1
O X X O O O X O X , 1
O X X O X O O O X , 1
O X X O O O X X O , 1
O X X O X X O O O , 1
O X X X O O X O O , 1
O X X O O O O X X , 1
X O X O O O X O X , 1

When I've got my program working, I'll consider applying it to this data. Thank you for providing it.

chrisw · Post by **chrisw** » Fri Sep 04, 2020 6:11 pm

towforce wrote: ↑Fri Sep 04, 2020 5:46 pm
chrisw wrote: ↑Fri Sep 04, 2020 5:36 pmWell, ignoring the agnorance, the challenge for you a few days ago was to "solve" tic-tac-toe before claiming 8*8 chess was "solvable" by polynomial. Here's the list of all possible unique tic-tac-toe positions as described, just to help you along. We'll await your polynomial expression with interest.

When I've got my program working, I'll consider applying it to this data. Thank you for providing it.

What you call "this data" is actually "the data", it's the game tree.

It shouldn't take long, here's some constants for NN no hidden layer, 18 inputs, one output, or linear polynomial as it otherwise called:

cases 767 errors 504, so not much traction using raw board inputs.

Code: Select all

[array([[-0.6886282 ],
       [-1.0524448 ],
       [-1.0499924 ],
       [-0.8408627 ],
       [-0.52842486],
       [-1.09617   ],
       [-0.73924524],
       [-0.8374172 ],
       [-0.98166645],
       [ 0.55151075],
       [ 0.51133204],
       [ 1.0839546 ],
       [ 1.439524  ],
       [ 1.8395087 ],
       [ 1.0986392 ],
       [ 1.8955249 ],
       [ 1.1994233 ],
       [ 1.6113871 ]], dtype=float32), array([-1.547183], dtype=float32)]

chrisw · Post by **chrisw** » Fri Sep 04, 2020 6:34 pm

towforce wrote: ↑Fri Sep 04, 2020 4:47 pm
chrisw wrote: ↑Fri Sep 04, 2020 11:52 am Following the discussions on linear evaluations to 'solve' chess, and the unanswered challenge to provide some data for even the simple game of noughts and crosses, or tic-tac-toe as it is known, I did some quick experiments (Python, Keras)

1. Build the entire game tree, evaluating each terminal node, and backing the result up the tree (minimax). This finds 549945 possible board positions, each one with a game result.

2. To make life easier for a linear polynomial to express the result from any position, those 549945 positions were split into groups based on number of men on the board, nine groups.

3. To make life even easier, each group was cut down to remove all symmetrical positions. For example, at ply one, there are 9 positions, but this reduces to 3 once symmetries are taken out. Ply 5 reduces from 15120 positions to 174 and so on.

print("Noughts and crosses")
print("===================")
print("total possible boards from start position 549945")

print("ply 1 total non-mirror boards 3 of 9")
print("ply 2 total non-mirror boards 12 of 72")
print("ply 3 total non-mirror boards 38 of 504")
print("ply 4 total non-mirror boards 108 of 3024")
print("ply 5 total non-mirror boards 174 of 15120")
print("ply 6 total non-mirror boards 204 of 54720")
print("ply 7 total non-mirror boards 154 of 148176")
print("ply 8 total non-mirror boards 59 of 200448")
print("ply 9 total non-mirror boards 15 of 127872")
print()

4. For each ply depth, a network was (relatively exhaustively) trained, with position as input (18 cells, first 9 set to 1 if 'O', second nine set to 1 if 'X', and game result either 0.0, 0.5 or 1.0

5. First results are for a network with no hidden layer, eg a linear polynomial. At nply=1, a network will produce constant results, of course, all ply one positions are draws. A linear polynomial can disentangle the 12 cases at ply 2, and the 15 cases at ply 9, but intermediate plies not.

print("Results, no hidden layer, 2x9 inputs")
print("====================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 6")
print("nply 4 cases 108 errors 22")
print("nply 5 cases 174 errors 98")
print("nply 6 cases 204 errors 78")
print("nply 7 cases 154 errors 85")
print("nply 8 cases 59 errors 36")
print("nply 9 cases 15 errors 0")
print()

6. Next, a hidden layer is added (9 neurons). This improves the results, but is still not able to resolve the all non-linearities.

# 18 x 9 x 1 lots of errors
print("Results, one hidden layer of 9, 2x9 inputs")
print("==========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 12")
print("nply 5 cases 174 errors 8")
print("nply 6 cases 204 errors 5")
print("nply 7 cases 154 errors 10")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

7. Increasing the hidden layer size gives better results, but still some errors. Possibly more runs or tweaking learning rate and so on might reduce errors to zero. I'ld posit that the non-linearities are relatively non-trivial, even for tic-tac-toe, if they need more hidden layer width to resolve.

# 18 x 18 x 1 still a few errors
print("Results, one hidden layer of 18, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 1")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

8. Doubling again the hidden layer size, but still some errors.

# 18 x 36 x 1 still a few errors
print("Results, one hidden layer of 36, 2x9 inputs")
print("===========================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 1")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")

9. Adding another hidden layer (8 neurons) and errors still there.

# still some errors with an added hidden layer
print("Results, two hidden layers of 18 x 8, 2x9 inputs")
print("======================================")
print("nply 1 cases 3 errors 0")
print("nply 2 cases 12 errors 0")
print("nply 3 cases 38 errors 0")
print("nply 4 cases 108 errors 0")
print("nply 5 cases 174 errors 0")
print("nply 6 cases 204 errors 2")
print("nply 7 cases 154 errors 0")
print("nply 8 cases 59 errors 0")
print("nply 9 cases 15 errors 0")
print()

10. Create non-linear inputs (NNUE style) and try that ....

Probably won't bother. If tac-tac-toe has non-trivial non-linearities, then the extension to the idea that a linear polynomial operating on (undefined, unknown) non-linear inputs could be created to solve the vastly more complex game of chess doesn't have much going for it.

Congratulations! You have just demonstrated that using NNs to evaluate complex game positions is a bad idea.

Had you fitted a polynomial to all the positions, the resulting EF would always give you the best answer.

Btw, a "linear polynomial" is a polynomial of degree 1. At degree 2 or above, it becomes non linear.

Some more help for you, first shot NN with a couple of hidden layers, 767 discrete positions, errors 13. Suggests errors could be got to zero and thus a "solution" with some more parameter tinkering. NN is 18 inputs as described, first hidden layer of 36, second hidden layer of 9 and an output layer of 1

Here's the weights with which the non-linearites of tic-tac-toe are mostly unravelled in this case. Good luck finding your "emergent patterns".

Code: Select all

[array([[-7.21870735e-02,  7.63205647e-01, -4.46224928e-01,
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         2.55288005e-01,  3.32829952e-01, -6.58195674e-01,
         1.16098857e+00,  3.98015112e-01,  2.47494504e-01]], dtype=float32), array([-0.06272185, -0.22166003,  0.08196726,  0.04592703,  0.38199273,
       -0.6262659 ,  0.32198212, -0.5331509 ,  0.19345208, -0.06356163,
        0.46799424, -0.09055065, -0.54665184,  0.08786053, -0.19329132,
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       -0.24768619,  0.3125676 , -0.34247407,  0.09707404, -0.18503076,
        0.15335773, -0.53044176,  0.1136632 ,  0.20711522, -0.11091267,
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         1.75902635e-01, -9.01612461e-01, -1.59820080e+00,
         1.94816053e-01,  8.42333436e-01, -4.92728911e-02],
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         8.63853693e-02, -1.69756675e+00, -1.29408881e-01,
        -5.60565352e-01,  8.54975507e-02,  5.60050309e-01],
       [-2.64237499e+00, -4.80458029e-02, -2.06384644e-01,
        -7.01535761e-01, -2.10875154e-01, -8.85891080e-01,
        -9.00584698e-01,  1.27352881e+00,  6.35671914e-01],
       [ 4.34905648e-01, -2.32142508e-01, -1.14630163e-01,
         2.16502473e-02,  7.86976457e-01, -7.11893797e-01,
         2.07923636e-01, -2.83802822e-02,  4.89864200e-01],
       [ 6.70742452e-01,  1.04387596e-01, -1.36422241e+00,
         3.27684373e-01,  1.61530066e+00,  9.75790441e-01,
         7.36138284e-01, -6.88607633e-01,  3.72200161e-01],
       [-4.00603116e-01, -5.03902316e-01,  6.36310399e-01,
         6.42415524e-01, -4.52179223e-01, -6.96410537e-01,
        -1.02800861e-01,  1.85072750e-01, -3.27607900e-01],
       [ 6.66975617e-01, -1.26258343e-01,  5.75085819e-01,
         6.45637035e-01, -4.41991895e-01,  7.25824296e-01,
        -6.51237965e-01,  2.48884320e-01, -3.06049287e-01],
       [ 5.85181415e-01,  2.53391206e-01, -4.79158849e-01,
        -7.57023394e-01,  1.20966390e-01,  5.96221447e-01,
        -2.52927750e-01,  1.25094008e+00, -8.27865064e-01],
       [ 1.10044770e-01, -1.09883040e-01,  8.62792492e-01,
         1.25394189e+00,  2.13946238e-01,  5.58375597e-01,
        -1.38619781e-01, -5.39468527e-01, -2.59057927e+00],
       [-6.03544056e-01, -3.75609189e-01,  2.31268525e-01,
         2.70072192e-01, -1.54635191e+00,  4.44554031e-01,
        -4.02013808e-01, -4.05475110e-01,  1.05622388e-01],
       [ 1.20966576e-01, -2.64484316e-01, -1.80770352e-01,
         1.17361441e-01,  3.84483725e-01, -1.61237597e-01,
         7.40598321e-01,  9.31271538e-02,  1.36672184e-01],
       [-9.39413726e-01,  1.89574674e-01, -1.06657529e+00,
        -9.87231806e-02,  3.25388968e-01, -1.22159588e+00,
         3.20433289e-01,  2.79660195e-01,  5.38842306e-02],
       [ 3.61356169e-01, -3.23852628e-01,  6.53567255e-01,
        -2.36882016e-01, -7.03389764e-01,  4.48072642e-01,
        -8.83939564e-01, -1.54124632e-01,  1.65163204e-01],
       [-1.92817852e-01, -2.25993693e-01, -2.15508640e-02,
         4.70456779e-01,  7.50122547e-01,  5.70182383e-01,
        -3.02210003e-01,  3.24922144e-01, -5.77221632e-01],
       [ 2.54836202e-01,  2.02777520e-01,  1.94092050e-01,
         8.07427704e-01,  1.22922696e-01, -1.06438136e+00,
         2.73717672e-01, -5.35612822e-01,  6.91881716e-01],
       [ 2.95592606e-01, -4.72999752e-01,  6.59206867e-01,
        -2.18877420e-01,  9.92582679e-01,  3.11064512e-01,
         2.29598284e-01,  5.13468921e-01,  1.23150364e-01],
       [ 2.46539097e-02,  8.23761076e-02, -8.00820664e-02,
         8.79520178e-01,  5.13451457e-01,  5.62470198e-01,
        -1.22118904e-03,  2.75631130e-01,  7.73800090e-02],
       [ 4.56842870e-01,  1.46464389e-02, -7.92990088e-01,
        -1.88213229e+00,  5.76673210e-01, -6.08039945e-02,
        -8.92243981e-02,  1.22693598e+00, -3.96792382e-01],
       [-5.42830944e-01,  1.63566738e-01,  3.00890476e-01,
         6.84571385e-01, -5.03659725e-01,  1.04041779e+00,
        -7.86364973e-02, -4.19630051e-01, -1.85114646e+00],
       [ 1.03192592e+00, -1.01894155e-01, -3.91575284e-02,
         7.63676286e-01,  5.12616575e-01,  1.06774822e-01,
        -2.04125667e+00, -1.61400735e+00, -5.71605384e-01],
       [-6.22181356e-01, -3.99810672e-01, -2.13070199e-01,
         7.23028481e-01,  1.43356279e-01,  5.90527654e-01,
         2.83321261e-01, -4.47331190e-01, -5.34217477e-01],
       [ 4.67079252e-01,  3.82857062e-02, -4.80982631e-01,
        -7.24662960e-01, -2.89423838e-02,  7.30585515e-01,
         4.54374254e-02,  1.03717971e+00,  4.60682571e-01],
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         7.41367996e-01,  2.74717122e-01, -6.74941242e-01,
        -1.96597362e+00, -7.70301580e-01, -1.84967697e-01],
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        -7.65594319e-02, -8.70503008e-01, -1.06571233e+00,
        -3.87937725e-02,  4.51751143e-01,  7.54596442e-02],
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        -3.53650153e-01,  6.64981008e-01, -3.21822874e-02,
        -8.36941004e-01,  1.30109799e+00, -1.02702129e+00],
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         1.47418827e-01,  3.40582043e-01, -4.32794809e-01],
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       [-0.01893299],
       [ 2.5551128 ],
       [-1.8111084 ],
       [-2.2513752 ],
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towforce · Post by **towforce** » Sat Nov 15, 2025 11:00 pm

chrisw wrote: ↑Fri Sep 04, 2020 5:36 pm...Here's the list of all possible unique tic-tac-toe positions as described, just to help you along...

Code: Select all

O . . . . . . . . , 0
. O . . . . . . . , 0
. . . . O . . . . , 0
O X . . . . . . . , -1
O . X . . . . . . , -1
O . . . X . . . . , 0
O . . . . X . . . , -1
O . . . . . . . X , -1
X O . . . . . . . , 0
. O . X . . . . . , -1
. O . . X . . . . , 0
. O . . . . X . . , -1
. O . . . . . X . , 0
X . . . O . . . . , 0
. X . . O . . . . , -1
O X O . . . . . . , 0
O X . O . . . . . , 1
O X . . O . . . . , 1
O X . . . O . . . , 0
O X . . . . O . . , 1
O X . . . . . O . , 0
O X . . . . . . O , 0
O O X . . . . . . , -1
O . X O . . . . . , 1
O . X . O . . . . , 0
O . X . . O . . . , 0
O . X . . . O . . , 1
O . X . . . . O . , 0
O . X . . . . . O , 1
O O . . X . . . . , 0
O . O . X . . . . , 0
O . . . X O . . . , 0
O . . . X . . . O , 0
O O . . . X . . . , -1
O . . O . X . . . , 0
O . . . O X . . . , 1
O . . . . X O . . , 1
O . . . . X . O . , 0
O O . . . . . . X , -1
O . . . O . . . X , 0
O . . . . O . . X , 0
X O . O . . . . . , 0
X O . . O . . . . , 0
X O . . . O . . . , -1
X O . . . . . O . , -1
. O . X O . . . . , 1
. O . X . O . . . , 0
. O . X . . . O . , -1
. O . O X . . . . , 0
. O . . X . . O . , -1
. O . . O . X . . , 0
. O . . . O X . . , -1
. O . . O . . X . , 0
O X O X . . . . . , -1
O X O . X . . . . , 0
O X O . . . X . . , -1
O X O . . . . X . , -1
O X X O . . . . . , -1
O X . O X . . . . , -1
O X . O . X . . . , -1
O X . O . . X . . , -1
O X . O . . . X . , -1
O X . O . . . . X , -1
O X X . O . . . . , -1
O X . X O . . . . , -1
O X . . O X . . . , -1
O X . . O . X . . , -1
O X . . O . . X . , -1
O X . . O . . . X , -1
O X X . . O . . . , -1
O X . X . O . . . , -1
O X . . X O . . . , 0
O X . . . O X . . , -1
O X . . . O . X . , -1
O X . . . O . . X , -1
O X X . . . O . . , -1
O X . . X . O . . , -1
O X . . . X O . . , -1
O X . . . . O X . , -1
O X . . . . O . X , -1
O X X . . . . O . , -1
O X . . X . . O . , -1
O X . . . X . O . , -1
O X . . . . X O . , 0
O X . . . . . O X , 0
O X X . . . . . O , -1
O X . X . . . . O , -1
O X . . X . . . O , 0
O X . . . X . . O , -1
O X . . . . X . O , -1
O X . . . . . X O , -1
O O X . X . . . . , 0
O O X . . X . . . , 1
O O X . . . X . . , -1
O O X . . . . X . , 0
O O X . . . . . X , 1
O . X O X . . . . , -1
O . X O . X . . . , -1
O . X O . . . X . , -1
O . X O . . . . X , -1
O . X . O X . . . , -1
O . X . O . X . . , -1
O . X . O . . X . , -1
O . X . O . . . X , 0
O . X . X O . . . , 0
O . X . . O X . . , -1
O . X . . O . X . , -1
O . X . . O . . X , -1
O . X . X . O . . , -1
O . X . . X O . . , -1
O . X . . . O . X , -1
O . X . X . . O . , -1
O . X . . X . O . , -1
O . X . . . . O X , 0
O . X . X . . . O , -1
O . X . . . X . O , -1
O O . . X X . . . , -1
O O . . X . . X . , -1
O O . . X . . . X , -1
O . O . X . . X . , -1
O . . . X O . X . , 0
O . . . X O . . X , 0
O O . . . X . X . , -1
O O . . . X . . X , -1
O . . O . X . . X , -1
O . . . O X . X . , -1
O . . . O X . . X , -1
O . . . . X . O X , -1
X O X O . . . . . , -1
X O . O X . . . . , 0
X O . O . X . . . , 0
X O . O . . . . X , -1
X O X . O . . . . , -1
X O . X O . . . . , -1
X O . . O X . . . , -1
X O . . O . X . . , -1
X O . . O . . X . , 0
X O . . O . . . X , -1
X O . X . O . . . , -1
X O . . X O . . . , 0
X O . . . O X . . , 1
X O . . . O . X . , 0
X O . . . O . . X , -1
X O X . . . . O . , -1
X O . X . . . O . , -1
X O . . X . . O . , 1
X O . . . X . O . , -1
X O . . . . X O . , -1
X O . . . . . O X , -1
. O . X O X . . . , -1
. O . X O . X . . , -1
. O . X O . . X . , -1
. O . X O . . . X , -1
. O . X X O . . . , -1
. O . X . O X . . , -1
. O . X . O . X . , -1
. O . X X . . O . , 1
. O . X . X . O . , -1
. O . O X . . . X , -1
. O . . O . X X . , 0
. O . . O . X . X , -1
O X O X O . . . . , 1
O X O X . O . . . , 0
O X O X . . O . . , -1
O X O X . . . O . , 0
O X O X . . . . O , 1
O X O O X . . . . , -1
O X O . X . O . . , -1
O X O . X . . O . , 0
O X O O . . X . . , -1
O X O . O . X . . , 0
O X O . . O X . . , 0
O X O . . . X O . , 0
O X O . . . X . O , 1
O X O O . . . X . , -1
O X O . O . . X . , 1
O X O . . . O X . , -1
O X X O O . . . . , 1
O X X O . O . . . , 1
O X X O . . O . . , 1
O X X O . . . O . , 1
O X X O . . . . O , 1
O X . O X O . . . , -1
O X . O X . O . . , 1
O X . O X . . O . , 0
O X . O X . . . O , -1
O X . O O X . . . , 1
O X . O . X O . . , 1
O X . O . X . O . , 0
O X . O . X . . O , 1
O X . O O . X . . , 1
O X . O . O X . . , -1
O X . O . . X O . , 0
O X . O . . X . O , -1
O X . O O . . X . , 1
O X . O . O . X . , -1
O X . O . . O X . , 1
O X . O . . . X O , -1
O X . O O . . . X , 1
O X . O . O . . X , 1
O X . O . . O . X , 1
O X . O . . . O X , 0
O X X . O O . . . , 1
O X X . O . O . . , 1
O X X . O . . O . , 0
O X X . O . . . O , 1
O X . X O O . . . , 0
O X . X O . . . O , 1
O X . . O X O . . , 1
O X . . O X . O . , 0
O X . . O X . . O , 1
O X . . O O X . . , 1
O X . . O . X O . , 0
O X . . O . X . O , 1
O X . . O O . X . , 1
O X . . O . O X . , 1
O X . . O . . X O , 1
O X . . O O . . X , 0
O X . . O . O . X , 1
O X . . O . . O X , 0
O X X . . O O . . , 1
O X X . . O . O . , 1
O X X . . O . . O , -1
O X . X . O . O . , 0
O X . X . O . . O , 1
O X . . X O O . . , -1
O X . . X O . O . , 0
O X . . X O . . O , -1
O X . . . O X O . , 0
O X . . . O X . O , 1
O X . . . O O X . , -1
O X . . . O O . X , 0
O X . . . O . O X , 0
O X X . . . O O . , 1
O X X . . . O . O , 1
O X . . X . O O . , 1
O X . . X . O . O , -1
O X . . . X O O . , 1
O X . . . X O . O , 1
O X . . . . O O X , 0
O X X . . . . O O , 1
O X . . X . . O O , 0
O X . . . . X O O , 0
O O X O X . . . . , -1
O O X . X O . . . , -1
O O X . X . O . . , 0
O O X . X . . O . , -1
O O X . X . . . O , -1
O O X O . X . . . , -1
O O X . O X . . . , -1
O O X . . X O . . , -1
O O X . . X . O . , -1
O O X O . . X . . , -1
O O X . O . X . . , 1
O O X . . O X . . , -1
O O X . . . X O . , -1
O O X . . . X . O , -1
O O X O . . . X . , -1
O O X . O . . X . , -1
O O X . . O . X . , -1
O O X O . . . . X , -1
O O X . O . . . X , -1
O O X . . O . . X , -1
O O X . . . O . X , -1
O O X . . . . O X , -1
O . X O X O . . . , -1
O . X O X . O . . , 1
O . X O X . . O . , -1
O . X O X . . . O , -1
O . X O O X . . . , -1
O . X O . X O . . , 1
O . X O . X . O . , -1
O . X O O . . X . , 1
O . X O . O . X . , 1
O . X O O . . . X , -1
O . X O . O . . X , 1
O . X O . . O . X , 1
O . X O . . . O X , -1
O . X . O X O . . , -1
O . X . O X . O . , -1
O . X . O O X . . , 1
O . X . O . X . O , 1
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X O . . . O X O . , -1
. O . X O X . O . , 1
. O . X O O X . . , -1
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. O . X X O . O . , -1
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O X X O X O . . . , -1
O X X O . O X . . , -1
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O X . O . O . X X , -1
O X X X O O . . . , -1
O X X . O O X . . , -1
O X X . O O . X . , -1
O X X . O O . . X , -1
O X X . O X O . . , -1
O X X . O . O X . , -1
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O X . . O O . X X , -1
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O X X X . O . O . , -1
O X X . X O . O . , -1
O X X . . O X O . , -1
O X X . . O . O X , -1
O X X X . O . . O , -1
O X X . X O . . O , 1
O X X . . O X . O , -1
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O X . X . O . O X , 0
O X . X X O . . O , -1
O X . X . O X . O , -1
O X . . X O O X . , 1
O X . . X O O . X , -1
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O X . . X O . O X , 0
O X . . X O X . O , -1
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O X X . X . O O . , -1
O X X . . X O O . , -1
O X X . . . O O X , -1
O X X . X . O . O , -1
O X X . . X O . O , -1
O X . . X X O O . , -1
O X . . X . O O X , -1
O X . . X X O . O , -1
O X . . . X O O X , -1
O X X . X . . O O , -1
O X X . . . X O O , -1
O X . . X . X O O , 0
O O X O X X . . . , -1
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O O X O X . . X . , -1
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O O X . X O X . . , 1
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O O X . X . X . O , 1
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O O X . . X O . X , 1
O O X . . X X O . , -1
O O X . . X . O X , 1
O O X O . . X . X , 1
O O X . O . X X . , -1
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O O X . . O X X . , 1
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O O X . . . X O X , -1
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O O X . O . . X X , 1
O O X . . O . X X , 0
O . X O X O . X . , -1
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O . X O X X . O . , -1
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O . X O O X . . X , 1
O . X O . X . O X , 1
O . X O O . . X X , -1
O . X O . O . X X , -1
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O . X . O X X O . , -1
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O . X . . O X O X , -1
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O O . O X X . X . , -1
O O . O X X . . X , -1
O O . . X X . O X , -1
O O . . X O . X X , -1
O O . O . X . X X , -1
O O . . O X . X X , -1
X O X O O X . . . , -1
X O X O O . X . . , -1
X O X O O . . X . , -1
X O X O O . . . X , -1
X O X O X O . . . , 1
X O X O . O X . . , -1
X O X O . O . X . , -1
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X O X O . X . O . , -1
X O X O . . . O X , -1
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X O . X O O X . . , 1
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X O . X O O . . X , -1
X O . . O O X X . , -1
X O . . O O X . X , -1
X O . X X O . O . , 1
X O . X . O X O . , 1
X O . . X O X O . , 1
. O . X O O X X . , 1
O X O X O X O . . , 1
O X O X O X . O . , 1
O X O X O O X . . , 0
O X O X O . X O . , 0
O X O X O . X . O , 1
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O X O X O O . . X , 0
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O X O X X O O . . , -1
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O X O X X O . . O , 1
O X O X . O X O . , 0
O X O X . O X . O , 1
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O X O X . O . O X , 0
O X O X X . O . O , -1
O X O X . X O . O , -1
O X O X X . . O O , -1
O X O X . . X O O , 1
O X O O X O X . . , -1
O X O O X . X O . , 0
O X O O X . X . O , -1
O X O O X . O . X , 1
O X O O X . . O X , 0
O X O . X . O O X , 0
O X O O O . X X . , -1
O X O O . O X X . , -1
O X O O . . X X O , -1
O X O O O . X . X , -1
O X O O . O X . X , -1
O X O O . . X O X , 0
O X O . O O X X . , -1
O X O . O . X X O , 1
O X O . O . X O X , 0
O X O . . O X X O , 1
O X X O O X O . . , 1
O X X O O X . O . , -1
O X X O O X . . O , 1
O X X O O O X . . , 1
O X X O O . X O . , 1
O X X O O . X . O , 1
O X X O O O . X . , 1
O X X O O . O X . , 1
O X X O O . . X O , 1
O X X O O O . . X , 1
O X X O O . O . X , 1
O X X O O . . O X , -1
O X X O X O O . . , 1
O X X O X O . O . , -1
O X X O X O . . O , -1
O X X O . O X O . , -1
O X X O . O X . O , -1
O X X O . O O X . , 1
O X X O . O . X O , -1
O X X O . O O . X , 1
O X X O . O . O X , 1
O X X O X . O O . , 1
O X X O X . . O O , -1
O X X O . X O O . , 1
O X X O . X . O O , 1
O X X O . . X O O , -1
O X X O . . O O X , 1
O X X O X . O . O , 1
O X X O . X O . O , 1
O X . O X O X O . , -1
O X . O X O X . O , -1
O X . O X O O . X , 1
O X . O X O . O X , 0
O X . O X X O O . , 1
O X . O X X . O O , 0
O X . O X . X O O , -1
O X . O X . O O X , 1
O X . O X X O . O , 1
O X . O O X X O . , 0
O X . O O X X . O , 1
O X . O O X O X . , 1
O X . O O X . X O , 1
O X . O O X O . X , 1
O X . O O X . O X , -1
O X . O . X X O O , 0
O X . O . X O O X , 1
O X . O O O X X . , 1
O X . O O . X X O , 1
O X . O O O X . X , 1
O X . O O . X O X , 0
O X . O . O X O X , 0
O X . O O O . X X , 1
O X X X O O . O . , 0
O X X X O O . . O , 1
O X X . O O X O . , 1
O X X . O O X . O , 1
O X X . O O O X . , 1
O X X . O O O . X , 0
O X X . O O . O X , 0
O X X . O X O O . , -1
O X X . O X O . O , 1
O X X . O . O O X , -1
O X X X O . . O O , 1
O X X . O . X O O , 1
O X . X O O . O X , 0
O X . . O X O O X , -1
O X . . O O X O X , 0
O X X . X O O O . , 1
O X X . X O O . O , -1
O X X . . O O O X , 0
O X X X . O . O O , 1
O X X . X O . O O , -1
O X X . . O X O O , -1
O X . X X O . O O , 1
O X . . X O O O X , 0
O X . . X O X O O , -1
O X X . X . O O O , 1
O O X O X X O . . , 1
O O X O X X . O . , -1
O O X O X O . X . , -1
O O X O X O . . X , -1
O O X O X . O . X , 1
O O X O X . . O X , -1
O O X . X O O . X , 0
O O X . X O . O X , -1
O O X . X X O O . , -1
O O X . X . O O X , -1
O O X O O X X . . , -1
O O X O . X X O . , -1
O O X O O X . X . , -1
O O X . O X X O . , 1
O O X O O . X . X , -1
O O X O . O X . X , -1
O O X . O O X X . , -1
O O X . O O X . X , -1
O O X . O . X O X , 1
O O X . . O X O X , -1
O O X O O . . X X , -1
O O X O . O . X X , -1
O O X . O O . X X , -1
O . X O X O O . X , 1
O . X O X O . O X , -1
O . X O X X O O . , 1
O . X O O O . X X , 1
O . X . O O X O X , 1
O O . O X X . O X , -1
O O . O O X . X X , -1
X O X O O X . O . , 1
X O X O O O X . . , 1
X O X O O . X O . , 1
X O X O O O . X . , 1
X O X O O . . O X , 1
X O X O X O . O . , -1
X O X O . O X O . , -1
X O . O O X . O X , 1
O X O X O X X O . , -1
O X O X O O X X . , -1
O X O X O O X . X , 0
O X O X O . X O X , 0
O X O X O O . X X , -1
O X O X X O O X . , 1
O X O X X O O . X , 0
O X O X X O X O . , -1
O X O X X O . O X , 0
O X O X . O X O X , 0
O X O X . O O X X , -1
O X O X X X O . O , 1
O X O X X . O X O , 1
O X O X . X O X O , -1
O X O X X . X O O , -1
O X O O X O X X . , 1
O X O O X O X . X , 0
O X O O X . X O X , 0
O X O O X . X X O , 1
O X O O O . X X X , 1
O X O O . O X X X , 1
O X X O O X X O . , -1
O X X O O X . O X , 1
O X X O O . X O X , -1
O X X O X O X O . , 1
O X X O X O . O X , -1
O X X O X O X . O , 1
O X X O X O . X O , 1
O X X O . O X O X , -1
O X X O . O X X O , -1
O X X O X X . O O , -1
O X X O X . X O O , 1
O X X O . X X O O , -1
O X . O X O X O X , 0
O X . O X X X O O , 0
O X . O O X X O X , 0
O X X X O O X O . , -1
O X X X O O . O X , 0
O X X . O O X O X , -1
O X X . O O O X X , -1
O X X . O X O O X , 1
O X X . X O O O X , -1
O X X X X O . O O , -1
O X X X . O X O O , -1
O X X . X O X O O , 1
O O X O X X X O . , 1
O O X O X X . O X , 1
O O X O X O . X X , -1
O O X O X O X . X , 1
O O X . X O X O X , 1
O O X . X X O O X , 1
O O X O O X X X . , -1
O O X O O X X . X , 1
O O X O . X X O X , 1
O O X O O X . X X , 1
O O X . O O X X X , 1
X O X O X O X O . , 1
X O X O X O . O X , 1
X O X O . O X O X , -1
O X O X O X X O O , 1
O X O X O O X X O , 1
O X O X O O X O X , 0
O X O X X O O O X , 0
O X O X X O X O O , 1
O X O X O X O X O , 1
O X O O X O X O X , 0
O X X O O X X O O , 1
O X X O O O X O X , 1
O X X O X O O O X , 1
O X X O O O X X O , 1
O X X O X X O O O , 1
O X X X O O X O O , 1
O X X O O O O X X , 1
X O X O O O X O X , 1

I've added line numbers for reference:

Code: Select all

1 O . . . . . . . . , 0
2 . O . . . . . . . , 0
3 . . . . O . . . . , 0
4 O X . . . . . . . , -1
5 O . X . . . . . . , -1
6 O . . . X . . . . , 0
7 O . . . . X . . . , -1
8 O . . . . . . . X , -1
9 X O . . . . . . . , 0
10 . O . X . . . . . , -1
11 . O . . X . . . . , 0
12 . O . . . . X . . , -1
13 . O . . . . . X . , 0
14 X . . . O . . . . , 0
15 . X . . O . . . . , -1
16 O X O . . . . . . , 0
17 O X . O . . . . . , 1
18 O X . . O . . . . , 1
19 O X . . . O . . . , 0
20 O X . . . . O . . , 1
21 O X . . . . . O . , 0
22 O X . . . . . . O , 0
23 O O X . . . . . . , -1
24 O . X O . . . . . , 1
25 O . X . O . . . . , 0
26 O . X . . O . . . , 0
27 O . X . . . O . . , 1
28 O . X . . . . O . , 0
29 O . X . . . . . O , 1
30 O O . . X . . . . , 0
31 O . O . X . . . . , 0
32 O . . . X O . . . , 0
33 O . . . X . . . O , 0
34 O O . . . X . . . , -1
35 O . . O . X . . . , 0
36 O . . . O X . . . , 1
37 O . . . . X O . . , 1
38 O . . . . X . O . , 0
39 O O . . . . . . X , -1
40 O . . . O . . . X , 0
41 O . . . . O . . X , 0
42 X O . O . . . . . , 0
43 X O . . O . . . . , 0
44 X O . . . O . . . , -1
45 X O . . . . . O . , -1
46 . O . X O . . . . , 1
47 . O . X . O . . . , 0
48 . O . X . . . O . , -1
49 . O . O X . . . . , 0
50 . O . . X . . O . , -1
51 . O . . O . X . . , 0
52 . O . . . O X . . , -1
53 . O . . O . . X . , 0
54 O X O X . . . . . , -1
55 O X O . X . . . . , 0
56 O X O . . . X . . , -1
57 O X O . . . . X . , -1
58 O X X O . . . . . , -1
59 O X . O X . . . . , -1
60 O X . O . X . . . , -1
61 O X . O . . X . . , -1
62 O X . O . . . X . , -1
63 O X . O . . . . X , -1
64 O X X . O . . . . , -1
65 O X . X O . . . . , -1
66 O X . . O X . . . , -1
67 O X . . O . X . . , -1
68 O X . . O . . X . , -1
69 O X . . O . . . X , -1
70 O X X . . O . . . , -1
71 O X . X . O . . . , -1
72 O X . . X O . . . , 0
73 O X . . . O X . . , -1
74 O X . . . O . X . , -1
75 O X . . . O . . X , -1
76 O X X . . . O . . , -1
77 O X . . X . O . . , -1
78 O X . . . X O . . , -1
79 O X . . . . O X . , -1
80 O X . . . . O . X , -1
81 O X X . . . . O . , -1
82 O X . . X . . O . , -1
83 O X . . . X . O . , -1
84 O X . . . . X O . , 0
85 O X . . . . . O X , 0
86 O X X . . . . . O , -1
87 O X . X . . . . O , -1
88 O X . . X . . . O , 0
89 O X . . . X . . O , -1
90 O X . . . . X . O , -1
91 O X . . . . . X O , -1
92 O O X . X . . . . , 0
93 O O X . . X . . . , 1
94 O O X . . . X . . , -1
95 O O X . . . . X . , 0
96 O O X . . . . . X , 1
97 O . X O X . . . . , -1
98 O . X O . X . . . , -1
99 O . X O . . . X . , -1
100 O . X O . . . . X , -1
101 O . X . O X . . . , -1
102 O . X . O . X . . , -1
103 O . X . O . . X . , -1
104 O . X . O . . . X , 0
105 O . X . X O . . . , 0
106 O . X . . O X . . , -1
107 O . X . . O . X . , -1
108 O . X . . O . . X , -1
109 O . X . X . O . . , -1
110 O . X . . X O . . , -1
111 O . X . . . O . X , -1
112 O . X . X . . O . , -1
113 O . X . . X . O . , -1
114 O . X . . . . O X , 0
115 O . X . X . . . O , -1
116 O . X . . . X . O , -1
117 O O . . X X . . . , -1
118 O O . . X . . X . , -1
119 O O . . X . . . X , -1
120 O . O . X . . X . , -1
121 O . . . X O . X . , 0
122 O . . . X O . . X , 0
123 O O . . . X . X . , -1
124 O O . . . X . . X , -1
125 O . . O . X . . X , -1
126 O . . . O X . X . , -1
127 O . . . O X . . X , -1
128 O . . . . X . O X , -1
129 X O X O . . . . . , -1
130 X O . O X . . . . , 0
131 X O . O . X . . . , 0
132 X O . O . . . . X , -1
133 X O X . O . . . . , -1
134 X O . X O . . . . , -1
135 X O . . O X . . . , -1
136 X O . . O . X . . , -1
137 X O . . O . . X . , 0
138 X O . . O . . . X , -1
139 X O . X . O . . . , -1
140 X O . . X O . . . , 0
141 X O . . . O X . . , 1
142 X O . . . O . X . , 0
143 X O . . . O . . X , -1
144 X O X . . . . O . , -1
145 X O . X . . . O . , -1
146 X O . . X . . O . , 1
147 X O . . . X . O . , -1
148 X O . . . . X O . , -1
149 X O . . . . . O X , -1
150 . O . X O X . . . , -1
151 . O . X O . X . . , -1
152 . O . X O . . X . , -1
153 . O . X O . . . X , -1
154 . O . X X O . . . , -1
155 . O . X . O X . . , -1
156 . O . X . O . X . , -1
157 . O . X X . . O . , 1
158 . O . X . X . O . , -1
159 . O . O X . . . X , -1
160 . O . . O . X X . , 0
161 . O . . O . X . X , -1
162 O X O X O . . . . , 1
163 O X O X . O . . . , 0
164 O X O X . . O . . , -1
165 O X O X . . . O . , 0
166 O X O X . . . . O , 1
167 O X O O X . . . . , -1
168 O X O . X . O . . , -1
169 O X O . X . . O . , 0
170 O X O O . . X . . , -1
171 O X O . O . X . . , 0
172 O X O . . O X . . , 0
173 O X O . . . X O . , 0
174 O X O . . . X . O , 1
175 O X O O . . . X . , -1
176 O X O . O . . X . , 1
177 O X O . . . O X . , -1
178 O X X O O . . . . , 1
179 O X X O . O . . . , 1
180 O X X O . . O . . , 1
181 O X X O . . . O . , 1
182 O X X O . . . . O , 1
183 O X . O X O . . . , -1
184 O X . O X . O . . , 1
185 O X . O X . . O . , 0
186 O X . O X . . . O , -1
187 O X . O O X . . . , 1
188 O X . O . X O . . , 1
189 O X . O . X . O . , 0
190 O X . O . X . . O , 1
191 O X . O O . X . . , 1
192 O X . O . O X . . , -1
193 O X . O . . X O . , 0
194 O X . O . . X . O , -1
195 O X . O O . . X . , 1
196 O X . O . O . X . , -1
197 O X . O . . O X . , 1
198 O X . O . . . X O , -1
199 O X . O O . . . X , 1
200 O X . O . O . . X , 1
201 O X . O . . O . X , 1
202 O X . O . . . O X , 0
203 O X X . O O . . . , 1
204 O X X . O . O . . , 1
205 O X X . O . . O . , 0
206 O X X . O . . . O , 1
207 O X . X O O . . . , 0
208 O X . X O . . . O , 1
209 O X . . O X O . . , 1
210 O X . . O X . O . , 0
211 O X . . O X . . O , 1
212 O X . . O O X . . , 1
213 O X . . O . X O . , 0
214 O X . . O . X . O , 1
215 O X . . O O . X . , 1
216 O X . . O . O X . , 1
217 O X . . O . . X O , 1
218 O X . . O O . . X , 0
219 O X . . O . O . X , 1
220 O X . . O . . O X , 0
221 O X X . . O O . . , 1
222 O X X . . O . O . , 1
223 O X X . . O . . O , -1
224 O X . X . O . O . , 0
225 O X . X . O . . O , 1
226 O X . . X O O . . , -1
227 O X . . X O . O . , 0
228 O X . . X O . . O , -1
229 O X . . . O X O . , 0
230 O X . . . O X . O , 1
231 O X . . . O O X . , -1
232 O X . . . O O . X , 0
233 O X . . . O . O X , 0
234 O X X . . . O O . , 1
235 O X X . . . O . O , 1
236 O X . . X . O O . , 1
237 O X . . X . O . O , -1
238 O X . . . X O O . , 1
239 O X . . . X O . O , 1
240 O X . . . . O O X , 0
241 O X X . . . . O O , 1
242 O X . . X . . O O , 0
243 O X . . . . X O O , 0
244 O O X O X . . . . , -1
245 O O X . X O . . . , -1
246 O O X . X . O . . , 0
247 O O X . X . . O . , -1
248 O O X . X . . . O , -1
249 O O X O . X . . . , -1
250 O O X . O X . . . , -1
251 O O X . . X O . . , -1
252 O O X . . X . O . , -1
253 O O X O . . X . . , -1
254 O O X . O . X . . , 1
255 O O X . . O X . . , -1
256 O O X . . . X O . , -1
257 O O X . . . X . O , -1
258 O O X O . . . X . , -1
259 O O X . O . . X . , -1
260 O O X . . O . X . , -1
261 O O X O . . . . X , -1
262 O O X . O . . . X , -1
263 O O X . . O . . X , -1
264 O O X . . . O . X , -1
265 O O X . . . . O X , -1
266 O . X O X O . . . , -1
267 O . X O X . O . . , 1
268 O . X O X . . O . , -1
269 O . X O X . . . O , -1
270 O . X O O X . . . , -1
271 O . X O . X O . . , 1
272 O . X O . X . O . , -1
273 O . X O O . . X . , 1
274 O . X O . O . X . , 1
275 O . X O O . . . X , -1
276 O . X O . O . . X , 1
277 O . X O . . O . X , 1
278 O . X O . . . O X , -1
279 O . X . O X O . . , -1
280 O . X . O X . O . , -1
281 O . X . O O X . . , 1
282 O . X . O . X . O , 1
283 O . X . O O . X . , 1
284 O . X . O O . . X , 0
285 O . X . O . O . X , -1
286 O . X . O . . O X , -1
287 O . X . X O O . . , 0
288 O . X . X O . O . , -1
289 O . X . . O X O . , -1
290 O . X . . O O . X , 0
291 O . X . . O . O X , 0
292 O . X . X . O O . , 1
293 O . X . X . O . O , 1
294 O . X . . X O O . , -1
295 O O . O X X . . . , 1
296 O O . . X X O . . , -1
297 O O . . X X . O . , -1
298 O O O . X . . X . , 1
299 O O . . X O . X . , 0
300 O O . O X . . . X , 1
301 O O . . X O . . X , 0
302 O O . . X . . O X , -1
303 O . . . X O . O X , 0
304 O O . O . X . X . , 1
305 O O . . O X . X . , 1
306 O O . O . X . . X , -1
307 O O . . O X . . X , -1
308 O O . . . X . O X , -1
309 O . . O O X . . X , -1
310 O . . O . X . O X , -1
311 O . . . O X . O X , -1
312 X O X O O . . . . , 1
313 X O X O . O . . . , -1
314 X O X O . . . O . , -1
315 X O . O X O . . . , -1
316 X O . O O X . . . , 0
317 X O . O . X . O . , 0
318 X O . O O . . . X , 1
319 X O . O . O . . X , -1
320 X O X . O . . O . , 1
321 X O . X O O . . . , -1
322 X O . X O . . O . , 1
323 X O . . O X . O . , 1
324 X O . . O O X . . , -1
325 X O . . O . X O . , 1
326 X O . . O O . X . , 0
327 X O . . O O . . X , 1
328 X O . . O . . O X , 1
329 X O . X . O . O . , -1
330 X O . . X O . O . , -1
331 X O . . . O X O . , -1
332 . O . X O X . O . , 1
333 . O . X O O X . . , -1
334 . O . X O O . X . , -1
335 . O . X X O . O . , -1
336 O X O X O X . . . , -1
337 O X O X O . X . . , -1
338 O X O X O . . X . , -1
339 O X O X O . . . X , -1
340 O X O X X O . . . , -1
341 O X O X . O X . . , -1
342 O X O X . O . X . , -1
343 O X O X . O . . X , 0
344 O X O X X . O . . , 1
345 O X O X . X O . . , -1
346 O X O X . . O . X , -1
347 O X O X X . . O . , 0
348 O X O X . X . O . , -1
349 O X O X . . X O . , -1
350 O X O X . . . O X , 0
351 O X O X X . . . O , -1
352 O X O X . . X . O , -1
353 O X O X . . . X O , -1
354 O X O O X . X . . , 0
355 O X O O X . . X . , 1
356 O X O O X . . . X , -1
357 O X O . X . O X . , 1
358 O X O . X . O . X , -1
359 O X O . X . X O . , 0
360 O X O O . . X X . , 1
361 O X O O . . X . X , 0
362 O X O . O . X X . , -1
363 O X O . O . X . X , 0
364 O X O . . O X X . , -1
365 O X O . . . X O X , 0
366 O X O . . . X X O , -1
367 O X X O O X . . . , -1
368 O X X O O . X . . , -1
369 O X X O O . . X . , -1
370 O X X O O . . . X , -1
371 O X X O X O . . . , -1
372 O X X O . O X . . , -1
373 O X X O . O . X . , -1
374 O X X O . O . . X , -1
375 O X X O X . . O . , -1
376 O X X O . X . O . , -1
377 O X X O . . X O . , -1
378 O X X O . . . O X , -1
379 O X X O X . . . O , -1
380 O X X O . X . . O , -1
381 O X X O . . X . O , -1
382 O X X O . . . X O , -1
383 O X . O X O X . . , 1
384 O X . O X O . X . , 1
385 O X . O X O . . X , -1
386 O X . O X X . O . , -1
387 O X . O X . X O . , 0
388 O X . O X . . O X , -1
389 O X . O X X . . O , -1
390 O X . O X . X . O , 1
391 O X . O X . . X O , 1
392 O X . O O X X . . , -1
393 O X . O O X . X . , -1
394 O X . O O X . . X , -1
395 O X . O . X X O . , 0
396 O X . O . X . O X , -1
397 O X . O . X X . O , -1
398 O X . O . X . X O , -1
399 O X . O O . X X . , -1
400 O X . O O . X . X , -1
401 O X . O . O X X . , -1
402 O X . O . O X . X , -1
403 O X . O . . X O X , 0
404 O X . O . . X X O , -1
405 O X . O O . . X X , -1
406 O X . O . O . X X , -1
407 O X X X O O . . . , -1
408 O X X . O O X . . , -1
409 O X X . O O . X . , -1
410 O X X . O O . . X , -1
411 O X X . O X O . . , -1
412 O X X . O . O X . , -1
413 O X X . O . O . X , -1
414 O X X X O . . O . , -1
415 O X X . O X . O . , -1
416 O X X . O . X O . , -1
417 O X X . O . . O X , 0
418 O X . X O O . X . , -1
419 O X . X O O . . X , 0
420 O X . . O X O X . , -1
421 O X . . O X O . X , -1
422 O X . . O X X O . , -1
423 O X . . O X . O X , 0
424 O X . . O O X X . , -1
425 O X . . O O X . X , -1
426 O X . . O . X O X , 0
427 O X . . O O . X X , -1
428 O X X . X O O . . , -1
429 O X X . . O O X . , -1
430 O X X . . O O . X , -1
431 O X X X . O . O . , -1
432 O X X . X O . O . , -1
433 O X X . . O X O . , -1
434 O X X . . O . O X , -1
435 O X X X . O . . O , -1
436 O X X . X O . . O , 1
437 O X X . . O X . O , -1
438 O X . X X O . O . , -1
439 O X . X . O . O X , 0
440 O X . X X O . . O , -1
441 O X . X . O X . O , -1
442 O X . . X O O X . , 1
443 O X . . X O O . X , -1
444 O X . . X O X O . , 0
445 O X . . X O . O X , 0
446 O X . . X O X . O , -1
447 O X . . . O X O X , 0
448 O X X . X . O O . , -1
449 O X X . . X O O . , -1
450 O X X . . . O O X , -1
451 O X X . X . O . O , -1
452 O X X . . X O . O , -1
453 O X . . X X O O . , -1
454 O X . . X . O O X , -1
455 O X . . X X O . O , -1
456 O X . . . X O O X , -1
457 O X X . X . . O O , -1
458 O X X . . . X O O , -1
459 O X . . X . X O O , 0
460 O O X O X X . . . , -1
461 O O X O X . X . . , 1
462 O O X O X . . X . , -1
463 O O X O X . . . X , -1
464 O O X . X O X . . , 1
465 O O X . X O . X . , 0
466 O O X . X O . . X , 0
467 O O X . X X O . . , -1
468 O O X . X . O . X , -1
469 O O X . X X . O . , 1
470 O O X . X . X O . , 1
471 O O X . X . . O X , 1
472 O O X . X . X . O , 1
473 O O X O . X X . . , 1
474 O O X O . X . X . , -1
475 O O X O . X . . X , 1
476 O O X . O X X . . , -1
477 O O X . O X . X . , -1
478 O O X . O X . . X , 1
479 O O X . . X O . X , 1
480 O O X . . X X O . , -1
481 O O X . . X . O X , 1
482 O O X O . . X . X , 1
483 O O X . O . X X . , -1
484 O O X . O . X . X , -1
485 O O X . . O X X . , 1
486 O O X . . O X . X , 1
487 O O X . . . X O X , -1
488 O O X O . . . X X , -1
489 O O X . O . . X X , 1
490 O O X . . O . X X , 0
491 O . X O X O . X . , -1
492 O . X O X O . . X , -1
493 O . X O X X . O . , -1
494 O . X O X . . O X , -1
495 O . X O O X . X . , -1
496 O . X O O X . . X , 1
497 O . X O . X . O X , 1
498 O . X O O . . X X , -1
499 O . X O . O . X X , -1
500 O . X . O X O . X , 1
501 O . X . O X X O . , -1
502 O . X . O X . O X , 1
503 O . X . O O X . X , -1
504 O . X . O O . X X , -1
505 O . X . X O O . X , -1
506 O . X . X O X O . , 1
507 O . X . X O . O X , 0
508 O . X . . O X O X , -1
509 O . X . X X O O . , -1
510 O O . O X X . X . , -1
511 O O . O X X . . X , -1
512 O O . . X X . O X , -1
513 O O . . X O . X X , -1
514 O O . O . X . X X , -1
515 O O . . O X . X X , -1
516 X O X O O X . . . , -1
517 X O X O O . X . . , -1
518 X O X O O . . X . , -1
519 X O X O O . . . X , -1
520 X O X O X O . . . , 1
521 X O X O . O X . . , -1
522 X O X O . O . X . , -1
523 X O X O X . . O . , 1
524 X O X O . X . O . , -1
525 X O X O . . . O X , -1
526 X O . O X O . X . , 0
527 X O . O X O . . X , 1
528 X O . O O X . X . , 0
529 X O . O O X . . X , -1
530 X O . O . X . O X , -1
531 X O . X O O X . . , 1
532 X O . X O O . X . , 0
533 X O . X O O . . X , -1
534 X O . . O O X X . , -1
535 X O . . O O X . X , -1
536 X O . X X O . O . , 1
537 X O . X . O X O . , 1
538 X O . . X O X O . , 1
539 . O . X O O X X . , 1
540 O X O X O X O . . , 1
541 O X O X O X . O . , 1
542 O X O X O O X . . , 0
543 O X O X O . X O . , 0
544 O X O X O . X . O , 1
545 O X O X O O . X . , 1
546 O X O X O . . X O , 1
547 O X O X O O . . X , 0
548 O X O X O . O . X , 1
549 O X O X O . . O X , 0
550 O X O X X O O . . , -1
551 O X O X X O . O . , 0
552 O X O X X O . . O , 1
553 O X O X . O X O . , 0
554 O X O X . O X . O , 1
555 O X O X . O O X . , -1
556 O X O X . O . X O , 1
557 O X O X . O O . X , 0
558 O X O X . O . O X , 0
559 O X O X X . O . O , -1
560 O X O X . X O . O , -1
561 O X O X X . . O O , -1
562 O X O X . . X O O , 1
563 O X O O X O X . . , -1
564 O X O O X . X O . , 0
565 O X O O X . X . O , -1
566 O X O O X . O . X , 1
567 O X O O X . . O X , 0
568 O X O . X . O O X , 0
569 O X O O O . X X . , -1
570 O X O O . O X X . , -1
571 O X O O . . X X O , -1
572 O X O O O . X . X , -1
573 O X O O . O X . X , -1
574 O X O O . . X O X , 0
575 O X O . O O X X . , -1
576 O X O . O . X X O , 1
577 O X O . O . X O X , 0
578 O X O . . O X X O , 1
579 O X X O O X O . . , 1
580 O X X O O X . O . , -1
581 O X X O O X . . O , 1
582 O X X O O O X . . , 1
583 O X X O O . X O . , 1
584 O X X O O . X . O , 1
585 O X X O O O . X . , 1
586 O X X O O . O X . , 1
587 O X X O O . . X O , 1
588 O X X O O O . . X , 1
589 O X X O O . O . X , 1
590 O X X O O . . O X , -1
591 O X X O X O O . . , 1
592 O X X O X O . O . , -1
593 O X X O X O . . O , -1
594 O X X O . O X O . , -1
595 O X X O . O X . O , -1
596 O X X O . O O X . , 1
597 O X X O . O . X O , -1
598 O X X O . O O . X , 1
599 O X X O . O . O X , 1
600 O X X O X . O O . , 1
601 O X X O X . . O O , -1
602 O X X O . X O O . , 1
603 O X X O . X . O O , 1
604 O X X O . . X O O , -1
605 O X X O . . O O X , 1
606 O X X O X . O . O , 1
607 O X X O . X O . O , 1
608 O X . O X O X O . , -1
609 O X . O X O X . O , -1
610 O X . O X O O . X , 1
611 O X . O X O . O X , 0
612 O X . O X X O O . , 1
613 O X . O X X . O O , 0
614 O X . O X . X O O , -1
615 O X . O X . O O X , 1
616 O X . O X X O . O , 1
617 O X . O O X X O . , 0
618 O X . O O X X . O , 1
619 O X . O O X O X . , 1
620 O X . O O X . X O , 1
621 O X . O O X O . X , 1
622 O X . O O X . O X , -1
623 O X . O . X X O O , 0
624 O X . O . X O O X , 1
625 O X . O O O X X . , 1
626 O X . O O . X X O , 1
627 O X . O O O X . X , 1
628 O X . O O . X O X , 0
629 O X . O . O X O X , 0
630 O X . O O O . X X , 1
631 O X X X O O . O . , 0
632 O X X X O O . . O , 1
633 O X X . O O X O . , 1
634 O X X . O O X . O , 1
635 O X X . O O O X . , 1
636 O X X . O O O . X , 0
637 O X X . O O . O X , 0
638 O X X . O X O O . , -1
639 O X X . O X O . O , 1
640 O X X . O . O O X , -1
641 O X X X O . . O O , 1
642 O X X . O . X O O , 1
643 O X . X O O . O X , 0
644 O X . . O X O O X , -1
645 O X . . O O X O X , 0
646 O X X . X O O O . , 1
647 O X X . X O O . O , -1
648 O X X . . O O O X , 0
649 O X X X . O . O O , 1
650 O X X . X O . O O , -1
651 O X X . . O X O O , -1
652 O X . X X O . O O , 1
653 O X . . X O O O X , 0
654 O X . . X O X O O , -1
655 O X X . X . O O O , 1
656 O O X O X X O . . , 1
657 O O X O X X . O . , -1
658 O O X O X O . X . , -1
659 O O X O X O . . X , -1
660 O O X O X . O . X , 1
661 O O X O X . . O X , -1
662 O O X . X O O . X , 0
663 O O X . X O . O X , -1
664 O O X . X X O O . , -1
665 O O X . X . O O X , -1
666 O O X O O X X . . , -1
667 O O X O . X X O . , -1
668 O O X O O X . X . , -1
669 O O X . O X X O . , 1
670 O O X O O . X . X , -1
671 O O X O . O X . X , -1
672 O O X . O O X X . , -1
673 O O X . O O X . X , -1
674 O O X . O . X O X , 1
675 O O X . . O X O X , -1
676 O O X O O . . X X , -1
677 O O X O . O . X X , -1
678 O O X . O O . X X , -1
679 O . X O X O O . X , 1
680 O . X O X O . O X , -1
681 O . X O X X O O . , 1
682 O . X O O O . X X , 1
683 O . X . O O X O X , 1
684 O O . O X X . O X , -1
685 O O . O O X . X X , -1
686 X O X O O X . O . , 1
687 X O X O O O X . . , 1
688 X O X O O . X O . , 1
689 X O X O O O . X . , 1
690 X O X O O . . O X , 1
691 X O X O X O . O . , -1
692 X O X O . O X O . , -1
693 X O . O O X . O X , 1
694 O X O X O X X O . , -1
695 O X O X O O X X . , -1
696 O X O X O O X . X , 0
697 O X O X O . X O X , 0
698 O X O X O O . X X , -1
699 O X O X X O O X . , 1
700 O X O X X O O . X , 0
701 O X O X X O X O . , -1
702 O X O X X O . O X , 0
703 O X O X . O X O X , 0
704 O X O X . O O X X , -1
705 O X O X X X O . O , 1
706 O X O X X . O X O , 1
707 O X O X . X O X O , -1
708 O X O X X . X O O , -1
709 O X O O X O X X . , 1
710 O X O O X O X . X , 0
711 O X O O X . X O X , 0
712 O X O O X . X X O , 1
713 O X O O O . X X X , 1
714 O X O O . O X X X , 1
715 O X X O O X X O . , -1
716 O X X O O X . O X , 1
717 O X X O O . X O X , -1
718 O X X O X O X O . , 1
719 O X X O X O . O X , -1
720 O X X O X O X . O , 1
721 O X X O X O . X O , 1
722 O X X O . O X O X , -1
723 O X X O . O X X O , -1
724 O X X O X X . O O , -1
725 O X X O X . X O O , 1
726 O X X O . X X O O , -1
727 O X . O X O X O X , 0
728 O X . O X X X O O , 0
729 O X . O O X X O X , 0
730 O X X X O O X O . , -1
731 O X X X O O . O X , 0
732 O X X . O O X O X , -1
733 O X X . O O O X X , -1
734 O X X . O X O O X , 1
735 O X X . X O O O X , -1
736 O X X X X O . O O , -1
737 O X X X . O X O O , -1
738 O X X . X O X O O , 1
739 O O X O X X X O . , 1
740 O O X O X X . O X , 1
741 O O X O X O . X X , -1
742 O O X O X O X . X , 1
743 O O X . X O X O X , 1
744 O O X . X X O O X , 1
745 O O X O O X X X . , -1
746 O O X O O X X . X , 1
747 O O X O . X X O X , 1
748 O O X O O X . X X , 1
749 O O X . O O X X X , 1
750 X O X O X O X O . , 1
751 X O X O X O . O X , 1
752 X O X O . O X O X , -1
753 O X O X O X X O O , 1
754 O X O X O O X X O , 1
755 O X O X O O X O X , 0
756 O X O X X O O O X , 0
757 O X O X X O X O O , 1
758 O X O X O X O X O , 1
759 O X O O X O X O X , 0
760 O X X O O X X O O , 1
761 O X X O O O X O X , 1
762 O X X O X O O O X , 1
763 O X X O O O X X O , 1
764 O X X O X X O O O , 1
765 O X X X O O X O O , 1
766 O X X O O O O X X , 1
767 X O X O O O X O X , 1

So here's what it means:

1. Game is noughts and crosses ("tic tac toe")

2. 0 goes first

3. Drawn positions evaluate to zero

4. Positions 0 wins evaluate to -1 (see row 4)

5 Positions X wins evaluate to 1 (see row 93)

Unfortunately, there are multiple errors in this data.

* Row 17: X is obliged to play bottom left to stop a row of 3. O then plays in the middle, forming two lines of two, thus winning the game. This should evaluate to -1, but the evaluation given is 1

* Rows 766 and 767: These are won by 0, because 0 has three in a row in the middle row, so again should evaluate to -1. However, the evaluation given is 1

This is saddening, because I was just about to have a go at it. Oh well - win some lose some...

chrisw · Post by **chrisw** » Sun Nov 16, 2025 12:32 am

towforce wrote: ↑Sat Nov 15, 2025 11:00 pm

chrisw wrote: ↑Fri Sep 04, 2020 5:36 pm...Here's the list of all possible unique tic-tac-toe positions as described, just to help you along...

Code: Select all

O . . . . . . . . , 0
. O . . . . . . . , 0
. . . . O . . . . , 0
O X . . . . . . . , -1
O . X . . . . . . , -1
O . . . X . . . . , 0
O . . . . X . . . , -1
O . . . . . . . X , -1
X O . . . . . . . , 0
. O . X . . . . . , -1
. O . . X . . . . , 0
. O . . . . X . . , -1
. O . . . . . X . , 0
X . . . O . . . . , 0
. X . . O . . . . , -1
O X O . . . . . . , 0
O X . O . . . . . , 1
O X . . O . . . . , 1
O X . . . O . . . , 0
O X . . . . O . . , 1
O X . . . . . O . , 0
O X . . . . . . O , 0
O O X . . . . . . , -1
O . X O . . . . . , 1
O . X . O . . . . , 0
O . X . . O . . . , 0
O . X . . . O . . , 1
O . X . . . . O . , 0
O . X . . . . . O , 1
O O . . X . . . . , 0
O . O . X . . . . , 0
O . . . X O . . . , 0
O . . . X . . . O , 0
O O . . . X . . . , -1
O . . O . X . . . , 0
O . . . O X . . . , 1
O . . . . X O . . , 1
O . . . . X . O . , 0
O O . . . . . . X , -1
O . . . O . . . X , 0
O . . . . O . . X , 0
X O . O . . . . . , 0
X O . . O . . . . , 0
X O . . . O . . . , -1
X O . . . . . O . , -1
. O . X O . . . . , 1
. O . X . O . . . , 0
. O . X . . . O . , -1
. O . O X . . . . , 0
. O . . X . . O . , -1
. O . . O . X . . , 0
. O . . . O X . . , -1
. O . . O . . X . , 0
O X O X . . . . . , -1
O X O . X . . . . , 0
O X O . . . X . . , -1
O X O . . . . X . , -1
O X X O . . . . . , -1
O X . O X . . . . , -1
O X . O . X . . . , -1
O X . O . . X . . , -1
O X . O . . . X . , -1
O X . O . . . . X , -1
O X X . O . . . . , -1
O X . X O . . . . , -1
O X . . O X . . . , -1
O X . . O . X . . , -1
O X . . O . . X . , -1
O X . . O . . . X , -1
O X X . . O . . . , -1
O X . X . O . . . , -1
O X . . X O . . . , 0
O X . . . O X . . , -1
O X . . . O . X . , -1
O X . . . O . . X , -1
O X X . . . O . . , -1
O X . . X . O . . , -1
O X . . . X O . . , -1
O X . . . . O X . , -1
O X . . . . O . X , -1
O X X . . . . O . , -1
O X . . X . . O . , -1
O X . . . X . O . , -1
O X . . . . X O . , 0
O X . . . . . O X , 0
O X X . . . . . O , -1
O X . X . . . . O , -1
O X . . X . . . O , 0
O X . . . X . . O , -1
O X . . . . X . O , -1
O X . . . . . X O , -1
O O X . X . . . . , 0
O O X . . X . . . , 1
O O X . . . X . . , -1
O O X . . . . X . , 0
O O X . . . . . X , 1
O . X O X . . . . , -1
O . X O . X . . . , -1
O . X O . . . X . , -1
O . X O . . . . X , -1
O . X . O X . . . , -1
O . X . O . X . . , -1
O . X . O . . X . , -1
O . X . O . . . X , 0
O . X . X O . . . , 0
O . X . . O X . . , -1
O . X . . O . X . , -1
O . X . . O . . X , -1
O . X . X . O . . , -1
O . X . . X O . . , -1
O . X . . . O . X , -1
O . X . X . . O . , -1
O . X . . X . O . , -1
O . X . . . . O X , 0
O . X . X . . . O , -1
O . X . . . X . O , -1
O O . . X X . . . , -1
O O . . X . . X . , -1
O O . . X . . . X , -1
O . O . X . . X . , -1
O . . . X O . X . , 0
O . . . X O . . X , 0
O O . . . X . X . , -1
O O . . . X . . X , -1
O . . O . X . . X , -1
O . . . O X . X . , -1
O . . . O X . . X , -1
O . . . . X . O X , -1
X O X O . . . . . , -1
X O . O X . . . . , 0
X O . O . X . . . , 0
X O . O . . . . X , -1
X O X . O . . . . , -1
X O . X O . . . . , -1
X O . . O X . . . , -1
X O . . O . X . . , -1
X O . . O . . X . , 0
X O . . O . . . X , -1
X O . X . O . . . , -1
X O . . X O . . . , 0
X O . . . O X . . , 1
X O . . . O . X . , 0
X O . . . O . . X , -1
X O X . . . . O . , -1
X O . X . . . O . , -1
X O . . X . . O . , 1
X O . . . X . O . , -1
X O . . . . X O . , -1
X O . . . . . O X , -1
. O . X O X . . . , -1
. O . X O . X . . , -1
. O . X O . . X . , -1
. O . X O . . . X , -1
. O . X X O . . . , -1
. O . X . O X . . , -1
. O . X . O . X . , -1
. O . X X . . O . , 1
. O . X . X . O . , -1
. O . O X . . . X , -1
. O . . O . X X . , 0
. O . . O . X . X , -1
O X O X O . . . . , 1
O X O X . O . . . , 0
O X O X . . O . . , -1
O X O X . . . O . , 0
O X O X . . . . O , 1
O X O O X . . . . , -1
O X O . X . O . . , -1
O X O . X . . O . , 0
O X O O . . X . . , -1
O X O . O . X . . , 0
O X O . . O X . . , 0
O X O . . . X O . , 0
O X O . . . X . O , 1
O X O O . . . X . , -1
O X O . O . . X . , 1
O X O . . . O X . , -1
O X X O O . . . . , 1
O X X O . O . . . , 1
O X X O . . O . . , 1
O X X O . . . O . , 1
O X X O . . . . O , 1
O X . O X O . . . , -1
O X . O X . O . . , 1
O X . O X . . O . , 0
O X . O X . . . O , -1
O X . O O X . . . , 1
O X . O . X O . . , 1
O X . O . X . O . , 0
O X . O . X . . O , 1
O X . O O . X . . , 1
O X . O . O X . . , -1
O X . O . . X O . , 0
O X . O . . X . O , -1
O X . O O . . X . , 1
O X . O . O . X . , -1
O X . O . . O X . , 1
O X . O . . . X O , -1
O X . O O . . . X , 1
O X . O . O . . X , 1
O X . O . . O . X , 1
O X . O . . . O X , 0
O X X . O O . . . , 1
O X X . O . O . . , 1
O X X . O . . O . , 0
O X X . O . . . O , 1
O X . X O O . . . , 0
O X . X O . . . O , 1
O X . . O X O . . , 1
O X . . O X . O . , 0
O X . . O X . . O , 1
O X . . O O X . . , 1
O X . . O . X O . , 0
O X . . O . X . O , 1
O X . . O O . X . , 1
O X . . O . O X . , 1
O X . . O . . X O , 1
O X . . O O . . X , 0
O X . . O . O . X , 1
O X . . O . . O X , 0
O X X . . O O . . , 1
O X X . . O . O . , 1
O X X . . O . . O , -1
O X . X . O . O . , 0
O X . X . O . . O , 1
O X . . X O O . . , -1
O X . . X O . O . , 0
O X . . X O . . O , -1
O X . . . O X O . , 0
O X . . . O X . O , 1
O X . . . O O X . , -1
O X . . . O O . X , 0
O X . . . O . O X , 0
O X X . . . O O . , 1
O X X . . . O . O , 1
O X . . X . O O . , 1
O X . . X . O . O , -1
O X . . . X O O . , 1
O X . . . X O . O , 1
O X . . . . O O X , 0
O X X . . . . O O , 1
O X . . X . . O O , 0
O X . . . . X O O , 0
O O X O X . . . . , -1
O O X . X O . . . , -1
O O X . X . O . . , 0
O O X . X . . O . , -1
O O X . X . . . O , -1
O O X O . X . . . , -1
O O X . O X . . . , -1
O O X . . X O . . , -1
O O X . . X . O . , -1
O O X O . . X . . , -1
O O X . O . X . . , 1
O O X . . O X . . , -1
O O X . . . X O . , -1
O O X . . . X . O , -1
O O X O . . . X . , -1
O O X . O . . X . , -1
O O X . . O . X . , -1
O O X O . . . . X , -1
O O X . O . . . X , -1
O O X . . O . . X , -1
O O X . . . O . X , -1
O O X . . . . O X , -1
O . X O X O . . . , -1
O . X O X . O . . , 1
O . X O X . . O . , -1
O . X O X . . . O , -1
O . X O O X . . . , -1
O . X O . X O . . , 1
O . X O . X . O . , -1
O . X O O . . X . , 1
O . X O . O . X . , 1
O . X O O . . . X , -1
O . X O . O . . X , 1
O . X O . . O . X , 1
O . X O . . . O X , -1
O . X . O X O . . , -1
O . X . O X . O . , -1
O . X . O O X . . , 1
O . X . O . X . O , 1
O . X . O O . X . , 1
O . X . O O . . X , 0
O . X . O . O . X , -1
O . X . O . . O X , -1
O . X . X O O . . , 0
O . X . X O . O . , -1
O . X . . O X O . , -1
O . X . . O O . X , 0
O . X . . O . O X , 0
O . X . X . O O . , 1
O . X . X . O . O , 1
O . X . . X O O . , -1
O O . O X X . . . , 1
O O . . X X O . . , -1
O O . . X X . O . , -1
O O O . X . . X . , 1
O O . . X O . X . , 0
O O . O X . . . X , 1
O O . . X O . . X , 0
O O . . X . . O X , -1
O . . . X O . O X , 0
O O . O . X . X . , 1
O O . . O X . X . , 1
O O . O . X . . X , -1
O O . . O X . . X , -1
O O . . . X . O X , -1
O . . O O X . . X , -1
O . . O . X . O X , -1
O . . . O X . O X , -1
X O X O O . . . . , 1
X O X O . O . . . , -1
X O X O . . . O . , -1
X O . O X O . . . , -1
X O . O O X . . . , 0
X O . O . X . O . , 0
X O . O O . . . X , 1
X O . O . O . . X , -1
X O X . O . . O . , 1
X O . X O O . . . , -1
X O . X O . . O . , 1
X O . . O X . O . , 1
X O . . O O X . . , -1
X O . . O . X O . , 1
X O . . O O . X . , 0
X O . . O O . . X , 1
X O . . O . . O X , 1
X O . X . O . O . , -1
X O . . X O . O . , -1
X O . . . O X O . , -1
. O . X O X . O . , 1
. O . X O O X . . , -1
. O . X O O . X . , -1
. O . X X O . O . , -1
O X O X O X . . . , -1
O X O X O . X . . , -1
O X O X O . . X . , -1
O X O X O . . . X , -1
O X O X X O . . . , -1
O X O X . O X . . , -1
O X O X . O . X . , -1
O X O X . O . . X , 0
O X O X X . O . . , 1
O X O X . X O . . , -1
O X O X . . O . X , -1
O X O X X . . O . , 0
O X O X . X . O . , -1
O X O X . . X O . , -1
O X O X . . . O X , 0
O X O X X . . . O , -1
O X O X . . X . O , -1
O X O X . . . X O , -1
O X O O X . X . . , 0
O X O O X . . X . , 1
O X O O X . . . X , -1
O X O . X . O X . , 1
O X O . X . O . X , -1
O X O . X . X O . , 0
O X O O . . X X . , 1
O X O O . . X . X , 0
O X O . O . X X . , -1
O X O . O . X . X , 0
O X O . . O X X . , -1
O X O . . . X O X , 0
O X O . . . X X O , -1
O X X O O X . . . , -1
O X X O O . X . . , -1
O X X O O . . X . , -1
O X X O O . . . X , -1
O X X O X O . . . , -1
O X X O . O X . . , -1
O X X O . O . X . , -1
O X X O . O . . X , -1
O X X O X . . O . , -1
O X X O . X . O . , -1
O X X O . . X O . , -1
O X X O . . . O X , -1
O X X O X . . . O , -1
O X X O . X . . O , -1
O X X O . . X . O , -1
O X X O . . . X O , -1
O X . O X O X . . , 1
O X . O X O . X . , 1
O X . O X O . . X , -1
O X . O X X . O . , -1
O X . O X . X O . , 0
O X . O X . . O X , -1
O X . O X X . . O , -1
O X . O X . X . O , 1
O X . O X . . X O , 1
O X . O O X X . . , -1
O X . O O X . X . , -1
O X . O O X . . X , -1
O X . O . X X O . , 0
O X . O . X . O X , -1
O X . O . X X . O , -1
O X . O . X . X O , -1
O X . O O . X X . , -1
O X . O O . X . X , -1
O X . O . O X X . , -1
O X . O . O X . X , -1
O X . O . . X O X , 0
O X . O . . X X O , -1
O X . O O . . X X , -1
O X . O . O . X X , -1
O X X X O O . . . , -1
O X X . O O X . . , -1
O X X . O O . X . , -1
O X X . O O . . X , -1
O X X . O X O . . , -1
O X X . O . O X . , -1
O X X . O . O . X , -1
O X X X O . . O . , -1
O X X . O X . O . , -1
O X X . O . X O . , -1
O X X . O . . O X , 0
O X . X O O . X . , -1
O X . X O O . . X , 0
O X . . O X O X . , -1
O X . . O X O . X , -1
O X . . O X X O . , -1
O X . . O X . O X , 0
O X . . O O X X . , -1
O X . . O O X . X , -1
O X . . O . X O X , 0
O X . . O O . X X , -1
O X X . X O O . . , -1
O X X . . O O X . , -1
O X X . . O O . X , -1
O X X X . O . O . , -1
O X X . X O . O . , -1
O X X . . O X O . , -1
O X X . . O . O X , -1
O X X X . O . . O , -1
O X X . X O . . O , 1
O X X . . O X . O , -1
O X . X X O . O . , -1
O X . X . O . O X , 0
O X . X X O . . O , -1
O X . X . O X . O , -1
O X . . X O O X . , 1
O X . . X O O . X , -1
O X . . X O X O . , 0
O X . . X O . O X , 0
O X . . X O X . O , -1
O X . . . O X O X , 0
O X X . X . O O . , -1
O X X . . X O O . , -1
O X X . . . O O X , -1
O X X . X . O . O , -1
O X X . . X O . O , -1
O X . . X X O O . , -1
O X . . X . O O X , -1
O X . . X X O . O , -1
O X . . . X O O X , -1
O X X . X . . O O , -1
O X X . . . X O O , -1
O X . . X . X O O , 0
O O X O X X . . . , -1
O O X O X . X . . , 1
O O X O X . . X . , -1
O O X O X . . . X , -1
O O X . X O X . . , 1
O O X . X O . X . , 0
O O X . X O . . X , 0
O O X . X X O . . , -1
O O X . X . O . X , -1
O O X . X X . O . , 1
O O X . X . X O . , 1
O O X . X . . O X , 1
O O X . X . X . O , 1
O O X O . X X . . , 1
O O X O . X . X . , -1
O O X O . X . . X , 1
O O X . O X X . . , -1
O O X . O X . X . , -1
O O X . O X . . X , 1
O O X . . X O . X , 1
O O X . . X X O . , -1
O O X . . X . O X , 1
O O X O . . X . X , 1
O O X . O . X X . , -1
O O X . O . X . X , -1
O O X . . O X X . , 1
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O O X . . . X O X , -1
O O X O . . . X X , -1
O O X . O . . X X , 1
O O X . . O . X X , 0
O . X O X O . X . , -1
O . X O X O . . X , -1
O . X O X X . O . , -1
O . X O X . . O X , -1
O . X O O X . X . , -1
O . X O O X . . X , 1
O . X O . X . O X , 1
O . X O O . . X X , -1
O . X O . O . X X , -1
O . X . O X O . X , 1
O . X . O X X O . , -1
O . X . O X . O X , 1
O . X . O O X . X , -1
O . X . O O . X X , -1
O . X . X O O . X , -1
O . X . X O X O . , 1
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O . X . . O X O X , -1
O . X . X X O O . , -1
O O . O X X . X . , -1
O O . O X X . . X , -1
O O . . X X . O X , -1
O O . . X O . X X , -1
O O . O . X . X X , -1
O O . . O X . X X , -1
X O X O O X . . . , -1
X O X O O . X . . , -1
X O X O O . . X . , -1
X O X O O . . . X , -1
X O X O X O . . . , 1
X O X O . O X . . , -1
X O X O . O . X . , -1
X O X O X . . O . , 1
X O X O . X . O . , -1
X O X O . . . O X , -1
X O . O X O . X . , 0
X O . O X O . . X , 1
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X O . O O X . . X , -1
X O . O . X . O X , -1
X O . X O O X . . , 1
X O . X O O . X . , 0
X O . X O O . . X , -1
X O . . O O X X . , -1
X O . . O O X . X , -1
X O . X X O . O . , 1
X O . X . O X O . , 1
X O . . X O X O . , 1
. O . X O O X X . , 1
O X O X O X O . . , 1
O X O X O X . O . , 1
O X O X O O X . . , 0
O X O X O . X O . , 0
O X O X O . X . O , 1
O X O X O O . X . , 1
O X O X O . . X O , 1
O X O X O O . . X , 0
O X O X O . O . X , 1
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O X O X X O O . . , -1
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O X O X X O . . O , 1
O X O X . O X O . , 0
O X O X . O X . O , 1
O X O X . O O X . , -1
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O X O X . O O . X , 0
O X O X . O . O X , 0
O X O X X . O . O , -1
O X O X . X O . O , -1
O X O X X . . O O , -1
O X O X . . X O O , 1
O X O O X O X . . , -1
O X O O X . X O . , 0
O X O O X . X . O , -1
O X O O X . O . X , 1
O X O O X . . O X , 0
O X O . X . O O X , 0
O X O O O . X X . , -1
O X O O . O X X . , -1
O X O O . . X X O , -1
O X O O O . X . X , -1
O X O O . O X . X , -1
O X O O . . X O X , 0
O X O . O O X X . , -1
O X O . O . X X O , 1
O X O . O . X O X , 0
O X O . . O X X O , 1
O X X O O X O . . , 1
O X X O O X . O . , -1
O X X O O X . . O , 1
O X X O O O X . . , 1
O X X O O . X O . , 1
O X X O O . X . O , 1
O X X O O O . X . , 1
O X X O O . O X . , 1
O X X O O . . X O , 1
O X X O O O . . X , 1
O X X O O . O . X , 1
O X X O O . . O X , -1
O X X O X O O . . , 1
O X X O X O . O . , -1
O X X O X O . . O , -1
O X X O . O X O . , -1
O X X O . O X . O , -1
O X X O . O O X . , 1
O X X O . O . X O , -1
O X X O . O O . X , 1
O X X O . O . O X , 1
O X X O X . O O . , 1
O X X O X . . O O , -1
O X X O . X O O . , 1
O X X O . X . O O , 1
O X X O . . X O O , -1
O X X O . . O O X , 1
O X X O X . O . O , 1
O X X O . X O . O , 1
O X . O X O X O . , -1
O X . O X O X . O , -1
O X . O X O O . X , 1
O X . O X O . O X , 0
O X . O X X O O . , 1
O X . O X X . O O , 0
O X . O X . X O O , -1
O X . O X . O O X , 1
O X . O X X O . O , 1
O X . O O X X O . , 0
O X . O O X X . O , 1
O X . O O X O X . , 1
O X . O O X . X O , 1
O X . O O X O . X , 1
O X . O O X . O X , -1
O X . O . X X O O , 0
O X . O . X O O X , 1
O X . O O O X X . , 1
O X . O O . X X O , 1
O X . O O O X . X , 1
O X . O O . X O X , 0
O X . O . O X O X , 0
O X . O O O . X X , 1
O X X X O O . O . , 0
O X X X O O . . O , 1
O X X . O O X O . , 1
O X X . O O X . O , 1
O X X . O O O X . , 1
O X X . O O O . X , 0
O X X . O O . O X , 0
O X X . O X O O . , -1
O X X . O X O . O , 1
O X X . O . O O X , -1
O X X X O . . O O , 1
O X X . O . X O O , 1
O X . X O O . O X , 0
O X . . O X O O X , -1
O X . . O O X O X , 0
O X X . X O O O . , 1
O X X . X O O . O , -1
O X X . . O O O X , 0
O X X X . O . O O , 1
O X X . X O . O O , -1
O X X . . O X O O , -1
O X . X X O . O O , 1
O X . . X O O O X , 0
O X . . X O X O O , -1
O X X . X . O O O , 1
O O X O X X O . . , 1
O O X O X X . O . , -1
O O X O X O . X . , -1
O O X O X O . . X , -1
O O X O X . O . X , 1
O O X O X . . O X , -1
O O X . X O O . X , 0
O O X . X O . O X , -1
O O X . X X O O . , -1
O O X . X . O O X , -1
O O X O O X X . . , -1
O O X O . X X O . , -1
O O X O O X . X . , -1
O O X . O X X O . , 1
O O X O O . X . X , -1
O O X O . O X . X , -1
O O X . O O X X . , -1
O O X . O O X . X , -1
O O X . O . X O X , 1
O O X . . O X O X , -1
O O X O O . . X X , -1
O O X O . O . X X , -1
O O X . O O . X X , -1
O . X O X O O . X , 1
O . X O X O . O X , -1
O . X O X X O O . , 1
O . X O O O . X X , 1
O . X . O O X O X , 1
O O . O X X . O X , -1
O O . O O X . X X , -1
X O X O O X . O . , 1
X O X O O O X . . , 1
X O X O O . X O . , 1
X O X O O O . X . , 1
X O X O O . . O X , 1
X O X O X O . O . , -1
X O X O . O X O . , -1
X O . O O X . O X , 1
O X O X O X X O . , -1
O X O X O O X X . , -1
O X O X O O X . X , 0
O X O X O . X O X , 0
O X O X O O . X X , -1
O X O X X O O X . , 1
O X O X X O O . X , 0
O X O X X O X O . , -1
O X O X X O . O X , 0
O X O X . O X O X , 0
O X O X . O O X X , -1
O X O X X X O . O , 1
O X O X X . O X O , 1
O X O X . X O X O , -1
O X O X X . X O O , -1
O X O O X O X X . , 1
O X O O X O X . X , 0
O X O O X . X O X , 0
O X O O X . X X O , 1
O X O O O . X X X , 1
O X O O . O X X X , 1
O X X O O X X O . , -1
O X X O O X . O X , 1
O X X O O . X O X , -1
O X X O X O X O . , 1
O X X O X O . O X , -1
O X X O X O X . O , 1
O X X O X O . X O , 1
O X X O . O X O X , -1
O X X O . O X X O , -1
O X X O X X . O O , -1
O X X O X . X O O , 1
O X X O . X X O O , -1
O X . O X O X O X , 0
O X . O X X X O O , 0
O X . O O X X O X , 0
O X X X O O X O . , -1
O X X X O O . O X , 0
O X X . O O X O X , -1
O X X . O O O X X , -1
O X X . O X O O X , 1
O X X . X O O O X , -1
O X X X X O . O O , -1
O X X X . O X O O , -1
O X X . X O X O O , 1
O O X O X X X O . , 1
O O X O X X . O X , 1
O O X O X O . X X , -1
O O X O X O X . X , 1
O O X . X O X O X , 1
O O X . X X O O X , 1
O O X O O X X X . , -1
O O X O O X X . X , 1
O O X O . X X O X , 1
O O X O O X . X X , 1
O O X . O O X X X , 1
X O X O X O X O . , 1
X O X O X O . O X , 1
X O X O . O X O X , -1
O X O X O X X O O , 1
O X O X O O X X O , 1
O X O X O O X O X , 0
O X O X X O O O X , 0
O X O X X O X O O , 1
O X O X O X O X O , 1
O X O O X O X O X , 0
O X X O O X X O O , 1
O X X O O O X O X , 1
O X X O X O O O X , 1
O X X O O O X X O , 1
O X X O X X O O O , 1
O X X X O O X O O , 1
O X X O O O O X X , 1
X O X O O O X O X , 1

I've added line numbers for reference:

Code: Select all

1 O . . . . . . . . , 0
2 . O . . . . . . . , 0
3 . . . . O . . . . , 0
4 O X . . . . . . . , -1
5 O . X . . . . . . , -1
6 O . . . X . . . . , 0
7 O . . . . X . . . , -1
8 O . . . . . . . X , -1
9 X O . . . . . . . , 0
10 . O . X . . . . . , -1
11 . O . . X . . . . , 0
12 . O . . . . X . . , -1
13 . O . . . . . X . , 0
14 X . . . O . . . . , 0
15 . X . . O . . . . , -1
16 O X O . . . . . . , 0
17 O X . O . . . . . , 1
18 O X . . O . . . . , 1
19 O X . . . O . . . , 0
20 O X . . . . O . . , 1
21 O X . . . . . O . , 0
22 O X . . . . . . O , 0
23 O O X . . . . . . , -1
24 O . X O . . . . . , 1
25 O . X . O . . . . , 0
26 O . X . . O . . . , 0
27 O . X . . . O . . , 1
28 O . X . . . . O . , 0
29 O . X . . . . . O , 1
30 O O . . X . . . . , 0
31 O . O . X . . . . , 0
32 O . . . X O . . . , 0
33 O . . . X . . . O , 0
34 O O . . . X . . . , -1
35 O . . O . X . . . , 0
36 O . . . O X . . . , 1
37 O . . . . X O . . , 1
38 O . . . . X . O . , 0
39 O O . . . . . . X , -1
40 O . . . O . . . X , 0
41 O . . . . O . . X , 0
42 X O . O . . . . . , 0
43 X O . . O . . . . , 0
44 X O . . . O . . . , -1
45 X O . . . . . O . , -1
46 . O . X O . . . . , 1
47 . O . X . O . . . , 0
48 . O . X . . . O . , -1
49 . O . O X . . . . , 0
50 . O . . X . . O . , -1
51 . O . . O . X . . , 0
52 . O . . . O X . . , -1
53 . O . . O . . X . , 0
54 O X O X . . . . . , -1
55 O X O . X . . . . , 0
56 O X O . . . X . . , -1
57 O X O . . . . X . , -1
58 O X X O . . . . . , -1
59 O X . O X . . . . , -1
60 O X . O . X . . . , -1
61 O X . O . . X . . , -1
62 O X . O . . . X . , -1
63 O X . O . . . . X , -1
64 O X X . O . . . . , -1
65 O X . X O . . . . , -1
66 O X . . O X . . . , -1
67 O X . . O . X . . , -1
68 O X . . O . . X . , -1
69 O X . . O . . . X , -1
70 O X X . . O . . . , -1
71 O X . X . O . . . , -1
72 O X . . X O . . . , 0
73 O X . . . O X . . , -1
74 O X . . . O . X . , -1
75 O X . . . O . . X , -1
76 O X X . . . O . . , -1
77 O X . . X . O . . , -1
78 O X . . . X O . . , -1
79 O X . . . . O X . , -1
80 O X . . . . O . X , -1
81 O X X . . . . O . , -1
82 O X . . X . . O . , -1
83 O X . . . X . O . , -1
84 O X . . . . X O . , 0
85 O X . . . . . O X , 0
86 O X X . . . . . O , -1
87 O X . X . . . . O , -1
88 O X . . X . . . O , 0
89 O X . . . X . . O , -1
90 O X . . . . X . O , -1
91 O X . . . . . X O , -1
92 O O X . X . . . . , 0
93 O O X . . X . . . , 1
94 O O X . . . X . . , -1
95 O O X . . . . X . , 0
96 O O X . . . . . X , 1
97 O . X O X . . . . , -1
98 O . X O . X . . . , -1
99 O . X O . . . X . , -1
100 O . X O . . . . X , -1
101 O . X . O X . . . , -1
102 O . X . O . X . . , -1
103 O . X . O . . X . , -1
104 O . X . O . . . X , 0
105 O . X . X O . . . , 0
106 O . X . . O X . . , -1
107 O . X . . O . X . , -1
108 O . X . . O . . X , -1
109 O . X . X . O . . , -1
110 O . X . . X O . . , -1
111 O . X . . . O . X , -1
112 O . X . X . . O . , -1
113 O . X . . X . O . , -1
114 O . X . . . . O X , 0
115 O . X . X . . . O , -1
116 O . X . . . X . O , -1
117 O O . . X X . . . , -1
118 O O . . X . . X . , -1
119 O O . . X . . . X , -1
120 O . O . X . . X . , -1
121 O . . . X O . X . , 0
122 O . . . X O . . X , 0
123 O O . . . X . X . , -1
124 O O . . . X . . X , -1
125 O . . O . X . . X , -1
126 O . . . O X . X . , -1
127 O . . . O X . . X , -1
128 O . . . . X . O X , -1
129 X O X O . . . . . , -1
130 X O . O X . . . . , 0
131 X O . O . X . . . , 0
132 X O . O . . . . X , -1
133 X O X . O . . . . , -1
134 X O . X O . . . . , -1
135 X O . . O X . . . , -1
136 X O . . O . X . . , -1
137 X O . . O . . X . , 0
138 X O . . O . . . X , -1
139 X O . X . O . . . , -1
140 X O . . X O . . . , 0
141 X O . . . O X . . , 1
142 X O . . . O . X . , 0
143 X O . . . O . . X , -1
144 X O X . . . . O . , -1
145 X O . X . . . O . , -1
146 X O . . X . . O . , 1
147 X O . . . X . O . , -1
148 X O . . . . X O . , -1
149 X O . . . . . O X , -1
150 . O . X O X . . . , -1
151 . O . X O . X . . , -1
152 . O . X O . . X . , -1
153 . O . X O . . . X , -1
154 . O . X X O . . . , -1
155 . O . X . O X . . , -1
156 . O . X . O . X . , -1
157 . O . X X . . O . , 1
158 . O . X . X . O . , -1
159 . O . O X . . . X , -1
160 . O . . O . X X . , 0
161 . O . . O . X . X , -1
162 O X O X O . . . . , 1
163 O X O X . O . . . , 0
164 O X O X . . O . . , -1
165 O X O X . . . O . , 0
166 O X O X . . . . O , 1
167 O X O O X . . . . , -1
168 O X O . X . O . . , -1
169 O X O . X . . O . , 0
170 O X O O . . X . . , -1
171 O X O . O . X . . , 0
172 O X O . . O X . . , 0
173 O X O . . . X O . , 0
174 O X O . . . X . O , 1
175 O X O O . . . X . , -1
176 O X O . O . . X . , 1
177 O X O . . . O X . , -1
178 O X X O O . . . . , 1
179 O X X O . O . . . , 1
180 O X X O . . O . . , 1
181 O X X O . . . O . , 1
182 O X X O . . . . O , 1
183 O X . O X O . . . , -1
184 O X . O X . O . . , 1
185 O X . O X . . O . , 0
186 O X . O X . . . O , -1
187 O X . O O X . . . , 1
188 O X . O . X O . . , 1
189 O X . O . X . O . , 0
190 O X . O . X . . O , 1
191 O X . O O . X . . , 1
192 O X . O . O X . . , -1
193 O X . O . . X O . , 0
194 O X . O . . X . O , -1
195 O X . O O . . X . , 1
196 O X . O . O . X . , -1
197 O X . O . . O X . , 1
198 O X . O . . . X O , -1
199 O X . O O . . . X , 1
200 O X . O . O . . X , 1
201 O X . O . . O . X , 1
202 O X . O . . . O X , 0
203 O X X . O O . . . , 1
204 O X X . O . O . . , 1
205 O X X . O . . O . , 0
206 O X X . O . . . O , 1
207 O X . X O O . . . , 0
208 O X . X O . . . O , 1
209 O X . . O X O . . , 1
210 O X . . O X . O . , 0
211 O X . . O X . . O , 1
212 O X . . O O X . . , 1
213 O X . . O . X O . , 0
214 O X . . O . X . O , 1
215 O X . . O O . X . , 1
216 O X . . O . O X . , 1
217 O X . . O . . X O , 1
218 O X . . O O . . X , 0
219 O X . . O . O . X , 1
220 O X . . O . . O X , 0
221 O X X . . O O . . , 1
222 O X X . . O . O . , 1
223 O X X . . O . . O , -1
224 O X . X . O . O . , 0
225 O X . X . O . . O , 1
226 O X . . X O O . . , -1
227 O X . . X O . O . , 0
228 O X . . X O . . O , -1
229 O X . . . O X O . , 0
230 O X . . . O X . O , 1
231 O X . . . O O X . , -1
232 O X . . . O O . X , 0
233 O X . . . O . O X , 0
234 O X X . . . O O . , 1
235 O X X . . . O . O , 1
236 O X . . X . O O . , 1
237 O X . . X . O . O , -1
238 O X . . . X O O . , 1
239 O X . . . X O . O , 1
240 O X . . . . O O X , 0
241 O X X . . . . O O , 1
242 O X . . X . . O O , 0
243 O X . . . . X O O , 0
244 O O X O X . . . . , -1
245 O O X . X O . . . , -1
246 O O X . X . O . . , 0
247 O O X . X . . O . , -1
248 O O X . X . . . O , -1
249 O O X O . X . . . , -1
250 O O X . O X . . . , -1
251 O O X . . X O . . , -1
252 O O X . . X . O . , -1
253 O O X O . . X . . , -1
254 O O X . O . X . . , 1
255 O O X . . O X . . , -1
256 O O X . . . X O . , -1
257 O O X . . . X . O , -1
258 O O X O . . . X . , -1
259 O O X . O . . X . , -1
260 O O X . . O . X . , -1
261 O O X O . . . . X , -1
262 O O X . O . . . X , -1
263 O O X . . O . . X , -1
264 O O X . . . O . X , -1
265 O O X . . . . O X , -1
266 O . X O X O . . . , -1
267 O . X O X . O . . , 1
268 O . X O X . . O . , -1
269 O . X O X . . . O , -1
270 O . X O O X . . . , -1
271 O . X O . X O . . , 1
272 O . X O . X . O . , -1
273 O . X O O . . X . , 1
274 O . X O . O . X . , 1
275 O . X O O . . . X , -1
276 O . X O . O . . X , 1
277 O . X O . . O . X , 1
278 O . X O . . . O X , -1
279 O . X . O X O . . , -1
280 O . X . O X . O . , -1
281 O . X . O O X . . , 1
282 O . X . O . X . O , 1
283 O . X . O O . X . , 1
284 O . X . O O . . X , 0
285 O . X . O . O . X , -1
286 O . X . O . . O X , -1
287 O . X . X O O . . , 0
288 O . X . X O . O . , -1
289 O . X . . O X O . , -1
290 O . X . . O O . X , 0
291 O . X . . O . O X , 0
292 O . X . X . O O . , 1
293 O . X . X . O . O , 1
294 O . X . . X O O . , -1
295 O O . O X X . . . , 1
296 O O . . X X O . . , -1
297 O O . . X X . O . , -1
298 O O O . X . . X . , 1
299 O O . . X O . X . , 0
300 O O . O X . . . X , 1
301 O O . . X O . . X , 0
302 O O . . X . . O X , -1
303 O . . . X O . O X , 0
304 O O . O . X . X . , 1
305 O O . . O X . X . , 1
306 O O . O . X . . X , -1
307 O O . . O X . . X , -1
308 O O . . . X . O X , -1
309 O . . O O X . . X , -1
310 O . . O . X . O X , -1
311 O . . . O X . O X , -1
312 X O X O O . . . . , 1
313 X O X O . O . . . , -1
314 X O X O . . . O . , -1
315 X O . O X O . . . , -1
316 X O . O O X . . . , 0
317 X O . O . X . O . , 0
318 X O . O O . . . X , 1
319 X O . O . O . . X , -1
320 X O X . O . . O . , 1
321 X O . X O O . . . , -1
322 X O . X O . . O . , 1
323 X O . . O X . O . , 1
324 X O . . O O X . . , -1
325 X O . . O . X O . , 1
326 X O . . O O . X . , 0
327 X O . . O O . . X , 1
328 X O . . O . . O X , 1
329 X O . X . O . O . , -1
330 X O . . X O . O . , -1
331 X O . . . O X O . , -1
332 . O . X O X . O . , 1
333 . O . X O O X . . , -1
334 . O . X O O . X . , -1
335 . O . X X O . O . , -1
336 O X O X O X . . . , -1
337 O X O X O . X . . , -1
338 O X O X O . . X . , -1
339 O X O X O . . . X , -1
340 O X O X X O . . . , -1
341 O X O X . O X . . , -1
342 O X O X . O . X . , -1
343 O X O X . O . . X , 0
344 O X O X X . O . . , 1
345 O X O X . X O . . , -1
346 O X O X . . O . X , -1
347 O X O X X . . O . , 0
348 O X O X . X . O . , -1
349 O X O X . . X O . , -1
350 O X O X . . . O X , 0
351 O X O X X . . . O , -1
352 O X O X . . X . O , -1
353 O X O X . . . X O , -1
354 O X O O X . X . . , 0
355 O X O O X . . X . , 1
356 O X O O X . . . X , -1
357 O X O . X . O X . , 1
358 O X O . X . O . X , -1
359 O X O . X . X O . , 0
360 O X O O . . X X . , 1
361 O X O O . . X . X , 0
362 O X O . O . X X . , -1
363 O X O . O . X . X , 0
364 O X O . . O X X . , -1
365 O X O . . . X O X , 0
366 O X O . . . X X O , -1
367 O X X O O X . . . , -1
368 O X X O O . X . . , -1
369 O X X O O . . X . , -1
370 O X X O O . . . X , -1
371 O X X O X O . . . , -1
372 O X X O . O X . . , -1
373 O X X O . O . X . , -1
374 O X X O . O . . X , -1
375 O X X O X . . O . , -1
376 O X X O . X . O . , -1
377 O X X O . . X O . , -1
378 O X X O . . . O X , -1
379 O X X O X . . . O , -1
380 O X X O . X . . O , -1
381 O X X O . . X . O , -1
382 O X X O . . . X O , -1
383 O X . O X O X . . , 1
384 O X . O X O . X . , 1
385 O X . O X O . . X , -1
386 O X . O X X . O . , -1
387 O X . O X . X O . , 0
388 O X . O X . . O X , -1
389 O X . O X X . . O , -1
390 O X . O X . X . O , 1
391 O X . O X . . X O , 1
392 O X . O O X X . . , -1
393 O X . O O X . X . , -1
394 O X . O O X . . X , -1
395 O X . O . X X O . , 0
396 O X . O . X . O X , -1
397 O X . O . X X . O , -1
398 O X . O . X . X O , -1
399 O X . O O . X X . , -1
400 O X . O O . X . X , -1
401 O X . O . O X X . , -1
402 O X . O . O X . X , -1
403 O X . O . . X O X , 0
404 O X . O . . X X O , -1
405 O X . O O . . X X , -1
406 O X . O . O . X X , -1
407 O X X X O O . . . , -1
408 O X X . O O X . . , -1
409 O X X . O O . X . , -1
410 O X X . O O . . X , -1
411 O X X . O X O . . , -1
412 O X X . O . O X . , -1
413 O X X . O . O . X , -1
414 O X X X O . . O . , -1
415 O X X . O X . O . , -1
416 O X X . O . X O . , -1
417 O X X . O . . O X , 0
418 O X . X O O . X . , -1
419 O X . X O O . . X , 0
420 O X . . O X O X . , -1
421 O X . . O X O . X , -1
422 O X . . O X X O . , -1
423 O X . . O X . O X , 0
424 O X . . O O X X . , -1
425 O X . . O O X . X , -1
426 O X . . O . X O X , 0
427 O X . . O O . X X , -1
428 O X X . X O O . . , -1
429 O X X . . O O X . , -1
430 O X X . . O O . X , -1
431 O X X X . O . O . , -1
432 O X X . X O . O . , -1
433 O X X . . O X O . , -1
434 O X X . . O . O X , -1
435 O X X X . O . . O , -1
436 O X X . X O . . O , 1
437 O X X . . O X . O , -1
438 O X . X X O . O . , -1
439 O X . X . O . O X , 0
440 O X . X X O . . O , -1
441 O X . X . O X . O , -1
442 O X . . X O O X . , 1
443 O X . . X O O . X , -1
444 O X . . X O X O . , 0
445 O X . . X O . O X , 0
446 O X . . X O X . O , -1
447 O X . . . O X O X , 0
448 O X X . X . O O . , -1
449 O X X . . X O O . , -1
450 O X X . . . O O X , -1
451 O X X . X . O . O , -1
452 O X X . . X O . O , -1
453 O X . . X X O O . , -1
454 O X . . X . O O X , -1
455 O X . . X X O . O , -1
456 O X . . . X O O X , -1
457 O X X . X . . O O , -1
458 O X X . . . X O O , -1
459 O X . . X . X O O , 0
460 O O X O X X . . . , -1
461 O O X O X . X . . , 1
462 O O X O X . . X . , -1
463 O O X O X . . . X , -1
464 O O X . X O X . . , 1
465 O O X . X O . X . , 0
466 O O X . X O . . X , 0
467 O O X . X X O . . , -1
468 O O X . X . O . X , -1
469 O O X . X X . O . , 1
470 O O X . X . X O . , 1
471 O O X . X . . O X , 1
472 O O X . X . X . O , 1
473 O O X O . X X . . , 1
474 O O X O . X . X . , -1
475 O O X O . X . . X , 1
476 O O X . O X X . . , -1
477 O O X . O X . X . , -1
478 O O X . O X . . X , 1
479 O O X . . X O . X , 1
480 O O X . . X X O . , -1
481 O O X . . X . O X , 1
482 O O X O . . X . X , 1
483 O O X . O . X X . , -1
484 O O X . O . X . X , -1
485 O O X . . O X X . , 1
486 O O X . . O X . X , 1
487 O O X . . . X O X , -1
488 O O X O . . . X X , -1
489 O O X . O . . X X , 1
490 O O X . . O . X X , 0
491 O . X O X O . X . , -1
492 O . X O X O . . X , -1
493 O . X O X X . O . , -1
494 O . X O X . . O X , -1
495 O . X O O X . X . , -1
496 O . X O O X . . X , 1
497 O . X O . X . O X , 1
498 O . X O O . . X X , -1
499 O . X O . O . X X , -1
500 O . X . O X O . X , 1
501 O . X . O X X O . , -1
502 O . X . O X . O X , 1
503 O . X . O O X . X , -1
504 O . X . O O . X X , -1
505 O . X . X O O . X , -1
506 O . X . X O X O . , 1
507 O . X . X O . O X , 0
508 O . X . . O X O X , -1
509 O . X . X X O O . , -1
510 O O . O X X . X . , -1
511 O O . O X X . . X , -1
512 O O . . X X . O X , -1
513 O O . . X O . X X , -1
514 O O . O . X . X X , -1
515 O O . . O X . X X , -1
516 X O X O O X . . . , -1
517 X O X O O . X . . , -1
518 X O X O O . . X . , -1
519 X O X O O . . . X , -1
520 X O X O X O . . . , 1
521 X O X O . O X . . , -1
522 X O X O . O . X . , -1
523 X O X O X . . O . , 1
524 X O X O . X . O . , -1
525 X O X O . . . O X , -1
526 X O . O X O . X . , 0
527 X O . O X O . . X , 1
528 X O . O O X . X . , 0
529 X O . O O X . . X , -1
530 X O . O . X . O X , -1
531 X O . X O O X . . , 1
532 X O . X O O . X . , 0
533 X O . X O O . . X , -1
534 X O . . O O X X . , -1
535 X O . . O O X . X , -1
536 X O . X X O . O . , 1
537 X O . X . O X O . , 1
538 X O . . X O X O . , 1
539 . O . X O O X X . , 1
540 O X O X O X O . . , 1
541 O X O X O X . O . , 1
542 O X O X O O X . . , 0
543 O X O X O . X O . , 0
544 O X O X O . X . O , 1
545 O X O X O O . X . , 1
546 O X O X O . . X O , 1
547 O X O X O O . . X , 0
548 O X O X O . O . X , 1
549 O X O X O . . O X , 0
550 O X O X X O O . . , -1
551 O X O X X O . O . , 0
552 O X O X X O . . O , 1
553 O X O X . O X O . , 0
554 O X O X . O X . O , 1
555 O X O X . O O X . , -1
556 O X O X . O . X O , 1
557 O X O X . O O . X , 0
558 O X O X . O . O X , 0
559 O X O X X . O . O , -1
560 O X O X . X O . O , -1
561 O X O X X . . O O , -1
562 O X O X . . X O O , 1
563 O X O O X O X . . , -1
564 O X O O X . X O . , 0
565 O X O O X . X . O , -1
566 O X O O X . O . X , 1
567 O X O O X . . O X , 0
568 O X O . X . O O X , 0
569 O X O O O . X X . , -1
570 O X O O . O X X . , -1
571 O X O O . . X X O , -1
572 O X O O O . X . X , -1
573 O X O O . O X . X , -1
574 O X O O . . X O X , 0
575 O X O . O O X X . , -1
576 O X O . O . X X O , 1
577 O X O . O . X O X , 0
578 O X O . . O X X O , 1
579 O X X O O X O . . , 1
580 O X X O O X . O . , -1
581 O X X O O X . . O , 1
582 O X X O O O X . . , 1
583 O X X O O . X O . , 1
584 O X X O O . X . O , 1
585 O X X O O O . X . , 1
586 O X X O O . O X . , 1
587 O X X O O . . X O , 1
588 O X X O O O . . X , 1
589 O X X O O . O . X , 1
590 O X X O O . . O X , -1
591 O X X O X O O . . , 1
592 O X X O X O . O . , -1
593 O X X O X O . . O , -1
594 O X X O . O X O . , -1
595 O X X O . O X . O , -1
596 O X X O . O O X . , 1
597 O X X O . O . X O , -1
598 O X X O . O O . X , 1
599 O X X O . O . O X , 1
600 O X X O X . O O . , 1
601 O X X O X . . O O , -1
602 O X X O . X O O . , 1
603 O X X O . X . O O , 1
604 O X X O . . X O O , -1
605 O X X O . . O O X , 1
606 O X X O X . O . O , 1
607 O X X O . X O . O , 1
608 O X . O X O X O . , -1
609 O X . O X O X . O , -1
610 O X . O X O O . X , 1
611 O X . O X O . O X , 0
612 O X . O X X O O . , 1
613 O X . O X X . O O , 0
614 O X . O X . X O O , -1
615 O X . O X . O O X , 1
616 O X . O X X O . O , 1
617 O X . O O X X O . , 0
618 O X . O O X X . O , 1
619 O X . O O X O X . , 1
620 O X . O O X . X O , 1
621 O X . O O X O . X , 1
622 O X . O O X . O X , -1
623 O X . O . X X O O , 0
624 O X . O . X O O X , 1
625 O X . O O O X X . , 1
626 O X . O O . X X O , 1
627 O X . O O O X . X , 1
628 O X . O O . X O X , 0
629 O X . O . O X O X , 0
630 O X . O O O . X X , 1
631 O X X X O O . O . , 0
632 O X X X O O . . O , 1
633 O X X . O O X O . , 1
634 O X X . O O X . O , 1
635 O X X . O O O X . , 1
636 O X X . O O O . X , 0
637 O X X . O O . O X , 0
638 O X X . O X O O . , -1
639 O X X . O X O . O , 1
640 O X X . O . O O X , -1
641 O X X X O . . O O , 1
642 O X X . O . X O O , 1
643 O X . X O O . O X , 0
644 O X . . O X O O X , -1
645 O X . . O O X O X , 0
646 O X X . X O O O . , 1
647 O X X . X O O . O , -1
648 O X X . . O O O X , 0
649 O X X X . O . O O , 1
650 O X X . X O . O O , -1
651 O X X . . O X O O , -1
652 O X . X X O . O O , 1
653 O X . . X O O O X , 0
654 O X . . X O X O O , -1
655 O X X . X . O O O , 1
656 O O X O X X O . . , 1
657 O O X O X X . O . , -1
658 O O X O X O . X . , -1
659 O O X O X O . . X , -1
660 O O X O X . O . X , 1
661 O O X O X . . O X , -1
662 O O X . X O O . X , 0
663 O O X . X O . O X , -1
664 O O X . X X O O . , -1
665 O O X . X . O O X , -1
666 O O X O O X X . . , -1
667 O O X O . X X O . , -1
668 O O X O O X . X . , -1
669 O O X . O X X O . , 1
670 O O X O O . X . X , -1
671 O O X O . O X . X , -1
672 O O X . O O X X . , -1
673 O O X . O O X . X , -1
674 O O X . O . X O X , 1
675 O O X . . O X O X , -1
676 O O X O O . . X X , -1
677 O O X O . O . X X , -1
678 O O X . O O . X X , -1
679 O . X O X O O . X , 1
680 O . X O X O . O X , -1
681 O . X O X X O O . , 1
682 O . X O O O . X X , 1
683 O . X . O O X O X , 1
684 O O . O X X . O X , -1
685 O O . O O X . X X , -1
686 X O X O O X . O . , 1
687 X O X O O O X . . , 1
688 X O X O O . X O . , 1
689 X O X O O O . X . , 1
690 X O X O O . . O X , 1
691 X O X O X O . O . , -1
692 X O X O . O X O . , -1
693 X O . O O X . O X , 1
694 O X O X O X X O . , -1
695 O X O X O O X X . , -1
696 O X O X O O X . X , 0
697 O X O X O . X O X , 0
698 O X O X O O . X X , -1
699 O X O X X O O X . , 1
700 O X O X X O O . X , 0
701 O X O X X O X O . , -1
702 O X O X X O . O X , 0
703 O X O X . O X O X , 0
704 O X O X . O O X X , -1
705 O X O X X X O . O , 1
706 O X O X X . O X O , 1
707 O X O X . X O X O , -1
708 O X O X X . X O O , -1
709 O X O O X O X X . , 1
710 O X O O X O X . X , 0
711 O X O O X . X O X , 0
712 O X O O X . X X O , 1
713 O X O O O . X X X , 1
714 O X O O . O X X X , 1
715 O X X O O X X O . , -1
716 O X X O O X . O X , 1
717 O X X O O . X O X , -1
718 O X X O X O X O . , 1
719 O X X O X O . O X , -1
720 O X X O X O X . O , 1
721 O X X O X O . X O , 1
722 O X X O . O X O X , -1
723 O X X O . O X X O , -1
724 O X X O X X . O O , -1
725 O X X O X . X O O , 1
726 O X X O . X X O O , -1
727 O X . O X O X O X , 0
728 O X . O X X X O O , 0
729 O X . O O X X O X , 0
730 O X X X O O X O . , -1
731 O X X X O O . O X , 0
732 O X X . O O X O X , -1
733 O X X . O O O X X , -1
734 O X X . O X O O X , 1
735 O X X . X O O O X , -1
736 O X X X X O . O O , -1
737 O X X X . O X O O , -1
738 O X X . X O X O O , 1
739 O O X O X X X O . , 1
740 O O X O X X . O X , 1
741 O O X O X O . X X , -1
742 O O X O X O X . X , 1
743 O O X . X O X O X , 1
744 O O X . X X O O X , 1
745 O O X O O X X X . , -1
746 O O X O O X X . X , 1
747 O O X O . X X O X , 1
748 O O X O O X . X X , 1
749 O O X . O O X X X , 1
750 X O X O X O X O . , 1
751 X O X O X O . O X , 1
752 X O X O . O X O X , -1
753 O X O X O X X O O , 1
754 O X O X O O X X O , 1
755 O X O X O O X O X , 0
756 O X O X X O O O X , 0
757 O X O X X O X O O , 1
758 O X O X O X O X O , 1
759 O X O O X O X O X , 0
760 O X X O O X X O O , 1
761 O X X O O O X O X , 1
762 O X X O X O O O X , 1
763 O X X O O O X X O , 1
764 O X X O X X O O O , 1
765 O X X X O O X O O , 1
766 O X X O O O O X X , 1
767 X O X O O O X O X , 1

So here's what it means:

1. Game is noughts and crosses ("tic tac toe")

2. 0 goes first

3. Drawn positions evaluate to zero

4. Positions 0 wins evaluate to -1 (see row 4)

5 Positions X wins evaluate to 1 (see row 93)

Unfortunately, there are multiple errors in this data.

* Row 17: X is obliged to play bottom left to stop a row of 3. O then plays in the middle, forming two lines of two, thus winning the game. This should evaluate to -1, but the evaluation given is 1

* Rows 766 and 767: These are won by 0, because 0 has three in a row in the middle row, so again should evaluate to -1. However, the evaluation given is 1

This is saddening, because I was just about to have a go at it. Oh well - win some lose some...

The guy who doesn’t understand minimax and negamax.

Dunning Kruger strikes again.

Linear evaluations with tic-tac-toe, some data

Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data

Re: Linear evaluations with tic-tac-toe, some data