Linear evaluations with tic-tac-toe, some data

[chrisw]

Do you need help to understand what you don’t understand? You revived a five year old thread misinterpreting negamax data whilst claiming you’ll show “deep patterns”, which I’m sure we are all looking forward to seeing after all these years of claims and NO data, NO evidence, just the same old woffle.

[Ajedrecista]

Hello Chris:

[...]

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()

[...]

Very interesting topic so far. Just a friendly awareness: I have found in several sites that the number of legal tic-tac-toe (or noughts and crosses) positions, up to rotation and reflection, is 765 and not 767, for example at OEIS as well as here and here. The different sources and you only disagree on 7-man (153 vs 154) and 8-man (57 vs 59), plus the trivial empty board, understandably not counted by you. Getting rid of the empty board, then there are 767 positions (yours) versus 764 (others), with an eye worth on 7 and 8-man positions to see what is really happening.

There are more tic-tac-toe related sequences at OEIS (link), from which I calculate some statistics of the 3×3 game. I am not sure, but I think that OEIS counts X as the first player and O as the second. If that:

Number of complete games of n X n tic-tac-toe, Number of complete games of n X n tic-tac-toe won by X, Number of complete games of n X n tic-tac-toe won by O and Number of complete games of n X n tic-tac-toe ending in a draw. From those, the expected score of the 3×3 game (n = 3 at OEIS) is:

Code: Select all

Score = 0   if the first player loses.
Score = 0.5 if players draw.
Score = 1   if the first player wins.

<Score> = (0*77904 + 0.5*46080 + 1*131184)/(77904 + 46080 + 131184)
<Score> = 154224/255168 = 1071/1772 ~ 0.6044

A shift from [0, 1] to [-1, 1] can be made. If I understood your posts correctly, -1 for you means a win for the first player (like my 1). Then:

Code: Select all

old values (mine)   in lowercase
NEW VALUES (CHRIS') IN UPPERCASE

min |→ MAX
max |→ MIN

SCORE = MIN + (MAX - MIN)*(max - score)/(max - min)
SCORE = -1 + [1 - (-1)]*(1 - 1071/1772)/(1 - 0)
SCORE = -185/886 ~ -0.2088

Number of tic-tac-toe games won after n plays and Number of complete games of n X n tic-tac-toe ending in a draw. From those, the expected number of plies of the 3×3 game (n = 3 at OEIS) is:

Code: Select all

<Plies> = [0*1 + 0*1 + 0*1 + 0*1 + 1440*5 + 5328*6 + 47952*7 + 72576*8 + (81792 + 46080)*9]/[0 + 0 + 0 + 0 + 1440 + 5328 + 47952 + 72576 + (81792 + 46080)]
<Plies> = 2106288/255168 = 14627/1772 ~ 8.2545

We all know that a perfect game is a draw (score = 0.5 or SCORE = 0) in 9 plies, of course. The former results should be valid for the full game tree and retrieve a substantial first-move advantage in long games, when they are random.

------------

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

Code: Select all

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Both of you agree up to 5-man (9, 72, 504, 3024 and 15120, when the first won games arrive) and disagree from there, but different things might be counted by each one, so this is not necessarily a mismatch.

------------

Lastly, I want to ask something because I am afraid I do not understand fully: the point of the thread is to say that increasing the layer sizes tend to reduce the number of errors (returned eval vs true outcome with best play), but it is not enough for completely getting rid of them, even in a simple game like tic-tac-toe? Thank you in advance.

Regards from Spain.

Ajedrecista.

[towforce]

The guy who doesn’t understand minimax and negamax.

Got it: the eval is for the player whose turn it is. Will take another look when time permits.

[towforce]

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Intuitively, the numbers generated by that expression look waaaaay too big.

[towforce]

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Intuitively, the numbers generated by that expression look waaaaay too big.

A quick look with a calculator enabled me to offer two simple proofs that the expression is wrong:

Proof 1

First go has 9 choices: Correct.

Second go has 8 choices, making the total choices 9*8 = 72: Correct

The mathematician has summed first and second go choices: Wrong


Proof 2

The number of choices for moves to turn 6 is 9*8*7*6*5*4 = 60480: Wrong: if the first player won on move 5, player 2 will not have a move

[Ajedrecista]

Hello Graham:

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Intuitively, the numbers generated by that expression look waaaaay too big.

The coefficients are 1, 9*8, 9*8*7, 9*8*7*6, 9*8*7*6*5, 9*8*7*6*5*4, 9*8*7*6*5*4*3, 9*8*7*6*5*4*3*2 and 9*8*7*6*5*4*3*2*1. I do not know if those numbers must be understood as exact values or upper bounds.

I pasted other link where, from 6-man, say 54528, 130464 and 142848, without giving a number for 9-man.

Regards from Spain.

Ajedrecista.

[towforce]

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Following the link, the above expression is for Ultimate Tic Tac Toe (link), not simple Tic Tac Toe.

It is actually a polynomial, with the x^n terms being monomials rather than turn indicators as I had thought. The expression as a polynomial would bear no relation to simple tic tac toe, except that the first few monomial coefficients match the actual number of move permutations in simple tic tac toe.

[chrisw]

Hello Chris:

[...]

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()

[...]

Very interesting topic so far. Just a friendly awareness: I have found in several sites that the number of legal tic-tac-toe (or noughts and crosses) positions, up to rotation and reflection, is 765 and not 767, for example at OEIS as well as here and here. The different sources and you only disagree on 7-man (153 vs 154) and 8-man (57 vs 59), plus the trivial empty board, understandably not counted by you. Getting rid of the empty board, then there are 767 positions (yours) versus 764 (others), with an eye worth on 7 and 8-man positions to see what is really happening.

There are more tic-tac-toe related sequences at OEIS (link), from which I calculate some statistics of the 3×3 game. I am not sure, but I think that OEIS counts X as the first player and O as the second. If that:

Number of complete games of n X n tic-tac-toe, Number of complete games of n X n tic-tac-toe won by X, Number of complete games of n X n tic-tac-toe won by O and Number of complete games of n X n tic-tac-toe ending in a draw. From those, the expected score of the 3×3 game (n = 3 at OEIS) is:

Code: Select all

Score = 0   if the first player loses.
Score = 0.5 if players draw.
Score = 1   if the first player wins.

<Score> = (0*77904 + 0.5*46080 + 1*131184)/(77904 + 46080 + 131184)
<Score> = 154224/255168 = 1071/1772 ~ 0.6044

A shift from [0, 1] to [-1, 1] can be made. If I understood your posts correctly, -1 for you means a win for the first player (like my 1). Then:

Code: Select all

old values (mine)   in lowercase
NEW VALUES (CHRIS') IN UPPERCASE

min |→ MAX
max |→ MIN

SCORE = MIN + (MAX - MIN)*(max - score)/(max - min)
SCORE = -1 + [1 - (-1)]*(1 - 1071/1772)/(1 - 0)
SCORE = -185/886 ~ -0.2088

Number of tic-tac-toe games won after n plays and Number of complete games of n X n tic-tac-toe ending in a draw. From those, the expected number of plies of the 3×3 game (n = 3 at OEIS) is:

Code: Select all

<Plies> = [0*1 + 0*1 + 0*1 + 0*1 + 1440*5 + 5328*6 + 47952*7 + 72576*8 + (81792 + 46080)*9]/[0 + 0 + 0 + 0 + 1440 + 5328 + 47952 + 72576 + (81792 + 46080)]
<Plies> = 2106288/255168 = 14627/1772 ~ 8.2545

We all know that a perfect game is a draw (score = 0.5 or SCORE = 0) in 9 plies, of course. The former results should be valid for the full game tree and retrieve a substantial first-move advantage in long games, when they are random.

------------

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

Code: Select all

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Both of you agree up to 5-man (9, 72, 504, 3024 and 15120, when the first won games arrive) and disagree from there, but different things might be counted by each one, so this is not necessarily a mismatch.

------------

Lastly, I want to ask something because I am afraid I do not understand fully: the point of the thread is to say that increasing the layer sizes tend to reduce the number of errors (returned eval vs true outcome with best play), but it is not enough for completely getting rid of them, even in a simple game like tic-tac-toe? Thank you in advance.

Regards from Spain.

Ajedrecista.

Bonjour Ajedrecista,

Last question first: the purpose of the thread was to pin down the towforce nonsense and endless repeated with zero data or evidence or argument that chess was solvable by linear polynomial. It’s not clear if he still asserts that exactly, since he now prefers to hide behind so called “deep patterns” which I guess he wants to plug into some function that then “solves” chess.

Since his deep patterns concept comes also with zero data or evidence, just endlessly repeating woffle, whilst being conveniently unfalsifiable (chess domain just too large), I figured to give him the much simpler and smaller tree, tic-tac-toe with all symmetries removed, and let him show “deep patterns”. That was five years ago, needless to say, he provided zilch, whilst simultaneously droning on about his alleged “deep patterns” in chess.

Towforce is no programmer, so not too easy for him, but the onus is on him to prove his assertions, the idea was to provide him with a simple and easy base set of data to work on. I’m not holding my breath, he prefers unverifiable and unfalsifiable woffle to any actual concrete science.

Hope that answers the question.

On the data - I prepped it five years ago, needless to say didn’t keep the sources, but it was an EPD-like list of all possible positions (minimax generated), which I then went through throwing out all rotations, reflections etc. Is the list two or three different from Reddit? Well, maybe, he can use the Reddit list if there is one.

On the network analysis:
I did a straight attempt at a linear polynomial (masses of PD code available for that). That failed, predictably. Data is non linear.

Next shot was to insert hidden layer(s). It’s known that NNs can handle non linearities using hidden layers. These worked (hidden layer = 1) and nearly reached perfection (hidden layers = 2). This shows (not proves) that the data is somehow consistent with some form of sensibility, and that it is non-linear.

Challenge to towforce is the STFU with the assertive zero content woffle and show where/how/why/what there are “deep patterns” to be extracted from the simpler domain of tic-tac-toe. It’s been made as simple as possible for him since five years ago. Don’t hold your breath.

[chrisw]

Hello Chris:

[...]



Lastly, I want to ask something because I am afraid I do not understand fully: the point of the thread is to say that increasing the layer sizes tend to reduce the number of errors (returned eval vs true outcome with best play), but it is not enough for completely getting rid of them, even in a simple game like tic-tac-toe? Thank you in advance.

Regards from Spain.

Ajedrecista.

The tree list is basically a EGTB lookup for the whole game, everything solved, so there's not much practical point in turning it into a NN lookup, save as an academic exercise. Clearly hidden layers are needed to deal with the non-linearities, and yes, the error rate from EGTB perfection seems to fall with increasing hidden layers, although I didn't go to any higher values for H. Error rate will probably also fall with one hidden layer of increasing width. One imagines that with a tail wind and the right phase of the moon and enough training there would be a "perfect" NN (zero errors) for such a simple problem, generatable from the perfect EGTB.

[Ajedrecista]

Hello:

Other source at Reddit gives a generating function of valid states at each move count (accounting for early wins), without further proofs:

1 + 9x + 72x² + 504x³ + 3024x⁴ + 15120x⁵ + 60480x⁶ + 181440x⁷ + 362880x⁸ + 362880x⁹

Following the link, the above expression is for Ultimate Tic Tac Toe (link), not simple Tic Tac Toe.

It is actually a polynomial, with the x^n terms being monomials rather than turn indicators as I had thought. The expression as a polynomial would bear no relation to simple tic tac toe, except that the first few monomial coefficients match the actual number of move permutations in simple tic tac toe.

@Graham: things cleared, there was a different version of the game. I wrote that 'different things might be counted by each one, so this is not necessarily a mismatch' because I was not sure at all about what that generation function was about.

I also wrote some coefficients from a link without giving the 9-man coefficient. It looks like I was looking at the wrong table. For 6-man to 9-man, there should be 54720, 148176, 200448 and 127872 for a total of 549946 with the empty board, exactly the Chris' result (empty board aside). The generating function of the possible paths should be:

Code: Select all

Paths(plies) = 1·(plies)⁰ + 9·(plies)¹ + 72·(plies)² + 504·(plies)³ + 3024·(plies)⁴ + 15120·(plies)⁵ + 54720·(plies)⁶ + 148176·(plies)⁷ + 200448·(plies)⁸ + 127872·(plies)⁹

Sum of all coefficients = 549946

------------

Bonjour Ajedrecista,

[...]

On the data - I prepped it five years ago, needless to say didn’t keep the sources, but it was an EPD-like list of all possible positions (minimax generated), which I then went through throwing out all rotations, reflections etc. Is the list two or three different from Reddit? Well, maybe, he can use the Reddit list if there is one.

[...]

@Chris: thank you for your answer. I took the number of different positions from OEIS and other sources. The Reddit post was other thing.

[...]

[...] Here's the list of all possible unique tic-tac-toe positions as described, [...]

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

Following 7-man and 8-man links, I got:

For the 7-man positions: there are +70=27-56 from first player POV versus +71=27-56 of yours, so the surplus position should be one evaluated as '1' in your list (from lines 540 to 693).

For the 8-man positions: there are +23=11-23 from first player POV versus +23=11-25 of yours, so the surplus positions should be two evaluated as '1' in your list (from lines 694 to 752).

Taking into account the purpose of the thread, identifying the surplus positions would take too much time to even consider seriously. I have been done more than enough.

Regards from Spain.

Ajedrecista.