I wrote a legality checker that gives the following percentages on a set of 10 million random positions:
59% Both Kings in Check, including 10% Adjacent Kings
17% Side not to move in Check
8% Unsupported Monochromatic Bishops Promotion
3% Illegal Triple Check
3% Illegal Double Check
---
90% definitely illegal
1% Discovered Check
5% Single Check
4% No Checks
---
10% possibly legal
Of the 1000 sample set, 96 are possibly legal, and of the 10,000 sample set, 942 are.
Both of these sets may be found at https://github.com/tromp/ChessPositionRanking,
I look forward to seeing other people run their legality checkers on these sets so we can compare results, and maybe track down bugs...
On the number of chess positions
Moderators: hgm, Rebel, chrisw
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tromp
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- Full name: John Tromp
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tromp
- Posts: 35
- Joined: Sun May 02, 2021 10:23 pm
- Full name: John Tromp
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tromp
- Posts: 35
- Joined: Sun May 02, 2021 10:23 pm
- Full name: John Tromp
Re: On the number of chess positions
Having completed the manual analysis of the 1000 random positions, I found 52 to be legal.
Three of these 52 positions occur at multiple ranks though. Two occur at 2 ranks, and one at 3 ranks.
So they account for a total of 56 occurrences in the set of all ranked positions, giving an average 56/52 = 1.077 number of occurences (slightly higher than the 1.033 avg over all positions in the sortedRnd1mFENs set).
We can thus estimate the number of legal positions as (52/1000) * 8726713169886222032347729969256422370854716254 / 1.077 = 4.2e44 +- 1.1e44, where the latter part reflects the 95% confidence bounds.
So we still have less than one digit of accuracy. But we can say the number is roughly 4e44...
A full analysis of the 10x larger sample at https://raw.githubusercontent.com/tromp ... Rnd10kFENs will give a significantly more accurate estimate, with confidence bounds +- 0.35e44
Three of these 52 positions occur at multiple ranks though. Two occur at 2 ranks, and one at 3 ranks.
So they account for a total of 56 occurrences in the set of all ranked positions, giving an average 56/52 = 1.077 number of occurences (slightly higher than the 1.033 avg over all positions in the sortedRnd1mFENs set).
We can thus estimate the number of legal positions as (52/1000) * 8726713169886222032347729969256422370854716254 / 1.077 = 4.2e44 +- 1.1e44, where the latter part reflects the 95% confidence bounds.
So we still have less than one digit of accuracy. But we can say the number is roughly 4e44...
A full analysis of the 10x larger sample at https://raw.githubusercontent.com/tromp ... Rnd10kFENs will give a significantly more accurate estimate, with confidence bounds +- 0.35e44
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tromp
- Posts: 35
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- Full name: John Tromp
Re: On the number of chess positions
All the software and a summary of this research project is now available at
https://github.com/tromp/ChessPositionRanking
including some bug bounties for finding mistakes.
I'd appreciate it if the chess programmers here could validate the double checking logic at
https://github.com/tromp/ChessPositionR ... hs#L74-L92
that tries to determine whether a double check is possible or not.
Happy bounty hunting!
https://github.com/tromp/ChessPositionRanking
including some bug bounties for finding mistakes.
I'd appreciate it if the chess programmers here could validate the double checking logic at
https://github.com/tromp/ChessPositionR ... hs#L74-L92
that tries to determine whether a double check is possible or not.
Happy bounty hunting!
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Ajedrecista
- Posts: 1966
- Joined: Wed Jul 13, 2011 9:04 pm
- Location: Madrid, Spain.
Re: On the number of chess positions.
Hello John:
So I would say that you are right: B+B, B+P, N+N, N+P, P+P, Q+Q and R+R are not able to give double checks. I spent some time with P+Q and P+R until I realised that they can give double check if the queen or rook is in the same column of the pawn and the enemy king and the pawn takes a piece giving check itself.
I post this reply for writing about B+N double checks and the enemy king trapped in a corner controlled by the bishop, where the side to move could define the legality or illegality of the position. First imagine this legal position:
[d]k7/p2K4/8/3N4/8/8/8/7B w - - 0 1
This is a checkmate in two moves starting with 1.- Nc7+ (double check with B+N) or 1.- Nb4+ (single check by B), then 1.- ..., Kb8; 2.- Na6#. Fine. With the white king on d8 instead of d7 (k2K4/p7/8/3N4/8/8/8/7B w - - 0 1), there are four checkmates in 2: the two solutions from before plus 1.- Nb6+ (double check with B+N) or 1.- Nf6+ (single check by B), then 1.- ..., Kb8; 2.- Nd7#. Also fine.
What about the white king on c8 instead of d7?
[d]k1K5/p7/8/3N4/8/8/8/7B w - - 0 1
Moving the knight to either square is checkmate, but... how the black king got trapped in this way? I can not find a legal way, do you? OTOH, with black to move:
[d]k1K5/p7/8/3N4/8/8/8/7B b - - 0 1
It looks legal to me, for example with a previous position with a black rook on c2 and the white king on d8, then 1.- ..., Rc8+; 2.- Kxd8.
[d]k2K4/p7/8/3N4/8/8/2r5/7B b - - 0 1
SUMMARY: all this bunch of positions is for thinking if there are positions (and how many positions) without any side in check, without adjacent kings and so on (apparently legal at a first glance) that can be reached legally or not depending on the side to move. What do you think on the matter?
------------
There are other more obvious illegal positions like:
- Three white pawns on g2, h2 and h3. At least one of these pawns is placed artificially.
- A bishop trapped by its on pawns (for example Bf1 with e2 and g2) and other white pawn at h2 or h3 without a white rook on g1, h1, h2 or h3 (how the rook flew?).
- Similar to the previous issue: white pieces Bf1, e2 and g2 with other white pawn at h2 or h3 and a black rook on g1, h1, h2 or h3 (how the black rook got there?).
- A bishop which should be trapped by its own pawns (like e2 and g2 before), yet still missing on f1 (how the white bishop flew?).
- Similar to the previous issue, a 'flying' bishop behind the enemy lines (white pawns at e2 and g2 and a black bishop at f1). How the black bishop got there?
- Similar to the previous issue: a white pawn on b2 and a black bishop on a1. How the black bishop got there?
I think you cope with these situations correctly, either automatically or manually.
------------
Good luck with your research! Please keep up your good work. I am following this thread with great interest.
Regards from Spain.
Ajedrecista.
I am not a programmer. Nevertheless, it seems that the couples of pieces with a value of 99 are not able to give double checks:tromp wrote: ↑Sun Aug 08, 2021 4:38 pm All the software and a summary of this research project is now available at
https://github.com/tromp/ChessPositionRanking
including some bug bounties for finding mistakes.
I'd appreciate it if the chess programmers here could validate the double checking logic at
https://github.com/tromp/ChessPositionR ... hs#L74-L92
that tries to determine whether a double check is possible or not.
Happy bounty hunting!
Code: Select all
minDist = M.fromList [("bb",99),("bn",7),("bp",99),("bq",5),("br",5),("nn",99),("np",99),
("nq",6),("nr",6),("pp",99),("pq",5),("pr",5),("qq",99),("qr",5),("rr",99)]I post this reply for writing about B+N double checks and the enemy king trapped in a corner controlled by the bishop, where the side to move could define the legality or illegality of the position. First imagine this legal position:
[d]k7/p2K4/8/3N4/8/8/8/7B w - - 0 1
This is a checkmate in two moves starting with 1.- Nc7+ (double check with B+N) or 1.- Nb4+ (single check by B), then 1.- ..., Kb8; 2.- Na6#. Fine. With the white king on d8 instead of d7 (k2K4/p7/8/3N4/8/8/8/7B w - - 0 1), there are four checkmates in 2: the two solutions from before plus 1.- Nb6+ (double check with B+N) or 1.- Nf6+ (single check by B), then 1.- ..., Kb8; 2.- Nd7#. Also fine.
What about the white king on c8 instead of d7?
[d]k1K5/p7/8/3N4/8/8/8/7B w - - 0 1
Moving the knight to either square is checkmate, but... how the black king got trapped in this way? I can not find a legal way, do you? OTOH, with black to move:
[d]k1K5/p7/8/3N4/8/8/8/7B b - - 0 1
It looks legal to me, for example with a previous position with a black rook on c2 and the white king on d8, then 1.- ..., Rc8+; 2.- Kxd8.
[d]k2K4/p7/8/3N4/8/8/2r5/7B b - - 0 1
SUMMARY: all this bunch of positions is for thinking if there are positions (and how many positions) without any side in check, without adjacent kings and so on (apparently legal at a first glance) that can be reached legally or not depending on the side to move. What do you think on the matter?
------------
There are other more obvious illegal positions like:
- Three white pawns on g2, h2 and h3. At least one of these pawns is placed artificially.
- A bishop trapped by its on pawns (for example Bf1 with e2 and g2) and other white pawn at h2 or h3 without a white rook on g1, h1, h2 or h3 (how the rook flew?).
- Similar to the previous issue: white pieces Bf1, e2 and g2 with other white pawn at h2 or h3 and a black rook on g1, h1, h2 or h3 (how the black rook got there?).
- A bishop which should be trapped by its own pawns (like e2 and g2 before), yet still missing on f1 (how the white bishop flew?).
- Similar to the previous issue, a 'flying' bishop behind the enemy lines (white pawns at e2 and g2 and a black bishop at f1). How the black bishop got there?
- Similar to the previous issue: a white pawn on b2 and a black bishop on a1. How the black bishop got there?
I think you cope with these situations correctly, either automatically or manually.
------------
Good luck with your research! Please keep up your good work. I am following this thread with great interest.
Regards from Spain.
Ajedrecista.
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Ajedrecista
- Posts: 1966
- Joined: Wed Jul 13, 2011 9:04 pm
- Location: Madrid, Spain.
Re: On the number of chess positions.
Hello:
- (Second example). The white rook could have been captured by a black knight, for example. A better definition could be if there are two white rooks outside that zone and all white pawns OTB, meaning no underpromotions to rooks, neither captures by black. Then, how the white rook flew? A variant could be two white rooks in g1 and h1, for example (how one of the white rooks got there?).
- (Third example). I would say it is illegal with the white h-pawn on h2. However, with the white h-pawn on h3, a black g-pawn can bypass the white shelter with captures: 1.- ... g3; 2.- Nh2 (from f3), gxh2; 3.- Rg1 (from h1), hxg1=R.
- (Fourth example). Similar to the second example, not only missing from f1 but standing elsewhere (for example on f3) with all the white pawns OTB, meaning no underpromotions to a light square bishop... how that white bishop flew? A variant could be more than one DSB or LSB per side with all the pawns of this side OTB, meaning no underpromotions to bishop.
- (Fifth example). 1.- f1=B coming from f2. If a white pawn is on f2, then the position would be illegal like in the second example. The black pawn advancing with captures would need f2 square empty.
- (Sixth example). Just like the fifth example, simply 1.- a1=B coming from a2 is enough. Things would be different (i.e. illegal) if we were talking about a white bishop (how the white bishop got there?) or with a white pawn on a2 (a similar theme that on the third example, where the difference was about a white pawn on h2 or h3 and a black pawn can bypass the white shelter with captures).
Regards from Spain.
Ajedrecista.
I have to correct myself. While I still see correct the first example, these are my thoughts with more time to think on the other examples:Ajedrecista wrote: ↑Wed Aug 11, 2021 8:39 pm[...]
There are other more obvious illegal positions like:
- Three white pawns on g2, h2 and h3. At least one of these pawns is placed artificially.
- A bishop trapped by its on pawns (for example Bf1 with e2 and g2) and other white pawn at h2 or h3 without a white rook on g1, h1, h2 or h3 (how the rook flew?).
- Similar to the previous issue: white pieces Bf1, e2 and g2 with other white pawn at h2 or h3 and a black rook on g1, h1, h2 or h3 (how the black rook got there?).
- A bishop which should be trapped by its own pawns (like e2 and g2 before), yet still missing on f1 (how the white bishop flew?).
- Similar to the previous issue, a 'flying' bishop behind the enemy lines (white pawns at e2 and g2 and a black bishop at f1). How the black bishop got there?
- Similar to the previous issue: a white pawn on b2 and a black bishop on a1. How the black bishop got there?
I think you cope with these situations correctly, either automatically or manually.
[...]
- (Second example). The white rook could have been captured by a black knight, for example. A better definition could be if there are two white rooks outside that zone and all white pawns OTB, meaning no underpromotions to rooks, neither captures by black. Then, how the white rook flew? A variant could be two white rooks in g1 and h1, for example (how one of the white rooks got there?).
- (Third example). I would say it is illegal with the white h-pawn on h2. However, with the white h-pawn on h3, a black g-pawn can bypass the white shelter with captures: 1.- ... g3; 2.- Nh2 (from f3), gxh2; 3.- Rg1 (from h1), hxg1=R.
- (Fourth example). Similar to the second example, not only missing from f1 but standing elsewhere (for example on f3) with all the white pawns OTB, meaning no underpromotions to a light square bishop... how that white bishop flew? A variant could be more than one DSB or LSB per side with all the pawns of this side OTB, meaning no underpromotions to bishop.
- (Fifth example). 1.- f1=B coming from f2. If a white pawn is on f2, then the position would be illegal like in the second example. The black pawn advancing with captures would need f2 square empty.
- (Sixth example). Just like the fifth example, simply 1.- a1=B coming from a2 is enough. Things would be different (i.e. illegal) if we were talking about a white bishop (how the white bishop got there?) or with a white pawn on a2 (a similar theme that on the third example, where the difference was about a white pawn on h2 or h3 and a black pawn can bypass the white shelter with captures).
Regards from Spain.
Ajedrecista.
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tromp
- Posts: 35
- Joined: Sun May 02, 2021 10:23 pm
- Full name: John Tromp
Re: On the number of chess positions.
Yes there are many non-check situations where legality depends on side to move.Ajedrecista wrote: ↑Wed Aug 11, 2021 8:39 pm SUMMARY: all this bunch of positions is for thinking if there are positions (and how many positions) without any side in check, without adjacent kings and so on (apparently legal at a first glance) that can be reached legally or not depending on the side to move. What do you think on the matter?
- Three white pawns on g2, h2 and h3. At least one of these pawns is placed artificially.
- A bishop trapped by its on pawns (for example Bf1 with e2 and g2) and other white pawn at h2 or h3 without a white rook on g1, h1, h2 or h3 (how the rook flew?).
- Similar to the previous issue: white pieces Bf1, e2 and g2 with other white pawn at h2 or h3 and a black rook on g1, h1, h2 or h3 (how the black rook got there?).
- A bishop which should be trapped by its own pawns (like e2 and g2 before), yet still missing on f1 (how the white bishop flew?).
- Similar to the previous issue, a 'flying' bishop behind the enemy lines (white pawns at e2 and g2 and a black bishop at f1). How the black bishop got there?
- Similar to the previous issue: a white pawn on b2 and a black bishop on a1. How the black bishop got there?
I think you cope with these situations correctly, either automatically or manually.
The most famous is the starting position, which cannot be legal with black to move.
Similarly, the position after 1. a3 is illegal with white to move.
Any such positions will have to be handled by manual analysis,
as do all the other cases you mention.
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tromp
- Posts: 35
- Joined: Sun May 02, 2021 10:23 pm
- Full name: John Tromp
Re: On the number of chess positions
I rewrote the README at https://github.com/tromp/ChessPositionRanking while adding lots of images.
A milestone has been set for obtaining a more accurate estimate by analyzing the 10k sample, which still has 895 unresolved issues.
Please consider volunteering to resolve at least one issue!
A milestone has been set for obtaining a more accurate estimate by analyzing the 10k sample, which still has 895 unresolved issues.
Please consider volunteering to resolve at least one issue!
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tromp
- Posts: 35
- Joined: Sun May 02, 2021 10:23 pm
- Full name: John Tromp
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Ajedrecista
- Posts: 1966
- Joined: Wed Jul 13, 2011 9:04 pm
- Location: Madrid, Spain.
Re: On the number of chess positions.
Hello John:
I compiled the FENs (without side to move) of 919 images of your GitHub (images and raw data) just for making things easier if anyone is interested here at TalkChess:
Regards from Spain.
Ajedrecista.
I wanted to write a very long post, but I must split it in shorter posts due to the 60000-character limitation of TalkChess.tromp wrote: ↑Sat Aug 14, 2021 11:28 pm I rewrote the README at https://github.com/tromp/ChessPositionRanking while adding lots of images.
A milestone has been set for obtaining a more accurate estimate by analyzing the 10k sample, which still has 895 unresolved issues.
Please consider volunteering to resolve at least one issue!
I compiled the FENs (without side to move) of 919 images of your GitHub (images and raw data) just for making things easier if anyone is interested here at TalkChess:
Code: Select all
3kB3/4pnb1/qpp2bR1/3Nr2r/NB2K1P1/nb1p1P2/nn3Br1/5B2
Nb3QRn/r3NN1B/3n1n1r/8/B2kbr2/1q2p1n1/p4p2/1RnbQNK1
RR4nQ/2q2K1p/1b4N1/3b1Rbn/pR3b2/1n1QB2k/B1n1rn2/4n1Q1
2nn2Nk/1RbbQQ2/3qp2b/1n1Nn3/1RrnK2n/n7/2NNr3/N3n1Q1
R3b1r1/Rrb3nR/2pk2Br/1bnr4/2bpqN2/2PB1N1q/6Q1/1Q2K1n1
R7/6n1/nQrQq3/2bBqqbb/QB1P1knq/q1Nr4/p5bR/R1K5
N1q1b3/1r2qnqR/6q1/k4qQ1/NR2R1nN/R3KP2/p1r4r/1B1r2r1
1NnB3n/b2rb3/2b2r1b/2N3p1/2Qp4/BQ1br3/K1PN1qkr/r1QB4
4N1q1/4rb1b/pnNRbQ2/rqrPQb2/1K1b2Q1/1n2b3/4Q2B/k2nQ3
q5b1/q1rR3R/2b2R1n/bQ1pp1nR/2R1Pnk1/5r2/NBK3b1/bQ2n3
3rb2Q/1B1b4/2n4P/BB2p1b1/1n1r2pk/RK4p1/1N1BbQp1/nq1Q1n2
Rb1Rn1b1/P5B1/1N3r2/Q1P2rr1/p2K1Q1q/N1q3p1/q2b3n/k2r2rN
1B2r3/1K2PR1R/Np1BP2b/6n1/1Brrn2p/1rqn1brp/2P3N1/kn6
Kb3b2/5pP1/P1R1Pnp1/n1n1Nrk1/1N2r3/1qb5/2b1rQn1/1b1Q1B1Q
1NRN4/1R1r3P/2n1n3/P2q3K/1b1qP3/7Q/1QnNkn1r/q1r1bb1n
1r1N1rq1/1R2NBp1/3bQ1P1/3Pn1Nq/4n1P1/P1pkb2q/K4n2/bb4n1
8/qrPk1Nbn/P2N1Pnb/3P2Nq/q1R5/1n2q1Qb/2K3b1/r2R1rb1
Q3Br2/4K3/N2B4/nnNqn2b/1b2p2R/2b2nn1/k2p2B1/qrQRn1QR
b3r3/2N2R2/B3K1kN/2Qpb2p/n1bp3b/2NN3r/1RbqR1p1/B1N2n2
r1b2Kq1/N1R5/2rkr1q1/r1q1r3/B1b2nrN/6B1/qN1n1Q1b/N1BR1N2
4n3/2K3bk/1R1r2qr/qB1Q2RB/BnRpNn1q/np3pNn/5B2/3b1B2
4B2n/Bqp2Nbr/r1r2p1B/5b2/n3bN2/1QR1q1b1/3kr2B/QKb2B1R
8/b1RNpBk1/2B3q1/1r2R1n1/3q1b2/1bRQn2Q/K2N3n/1nBn1Rnr
1qR1b2R/4Q1Q1/2Q1nN2/1r1b2NB/3p4/bpb5/bBBq2k1/r1BK2nb
1kbQ1Qr1/1q1QR3/r1rnqrR1/1n1q2rr/1b3NKB/r7/5qB1/2Q1B1N1
3nN3/bQR4q/q1qB2rn/4K1N1/3NpQq1/1b3Br1/B2n4/1BRbbkq1
4N2N/B3pn1N/1Nr5/pR1rb2k/3bqqn1/QNb2Qn1/2KQ2np/5B2
2N3bR/Q2Qp2B/2n1Q1R1/nnBnb2B/6qK/2qR2Bb/rq2k1r1/2r5
1RR5/r2b2bK/3B1b2/BN5P/2qnq1k1/1rqN1b2/B1NRp1rQ/2b1n2b
1B1qBR1b/4k2b/1q1nn2p/1Bb3rN/rbP1Q2b/2Nr4/K2n4/1Rb3QN
nBB1K3/1NrNrR1n/b1kr2Q1/bR6/1q1q1br1/7n/1p4Pp/r1B2RQ1
2Q2R1N/1rB2Bb1/pp6/1N3PrN/Q1n5/rn1rQ3/2bRKnbq/k1qb4
nN1qB1r1/B5k1/q1p5/1RRQ2r1/p3q1rN/bBn1P3/2pqQ3/qQ1b3K
5nBr/3Bb1pQ/2Rp1q1B/bnpQ3P/1kr1b3/R1rR2K1/q2bn1N1/4N3
bQ1RK2b/5nb1/7r/2B3kp/p1nRR3/bN1nPp2/B2QQ1nb/1r3qN1
nR1n3r/5r2/B1r4q/rq1QBR1q/b3B1B1/5Kn1/bPR2r2/1k2Nb1N
3b3b/1RB4B/r1R4B/2bR1NPr/2Rn1nqq/N1Qp1r1b/5q1k/1n2K2b
qn3rB1/2K1p3/qnqR1N1B/NB2Q2N/b6R/rrN2P2/1n2b1kb/2b2n2
RB2r1B1/rr1R2bq/Rb2p2n/p1n1P1Q1/2N1kb2/Kq4q1/BN2b3/5Bn1
B2Q4/2q1B3/br1n2kP/B4np1/1nRpRb2/1rq2N2/Rb2nb1B/1Kr1N3
bk1Nn3/1br4q/1R1P2b1/1q1nB2r/3rBQR1/5nqb/B2qB3/KQb3N1
q2b1Qq1/1nr1BrbR/1K6/3rB1qb/2B2p1b/NP2nBk1/1QNpb3/2R5
3n4/1r1p3B/p3pK2/BRrqq2N/nB2Nn2/2b3Q1/Pn2r1b1/RB1k1n1Q
3bqq2/1p6/3qBRN1/Q3pq1k/bR2P1r1/1r1nNpb1/1BK1QbNn/3N4
2rB2Nb/1Qb3p1/1N1R1p2/2n1bp1Q/1KP3R1/2p1pnRR/8/r1k1rBqb
r2QBr1r/1K1Qr3/n1Rr2p1/8/b2B1Qq1/6b1/k2BNpP1/qbnn1RR1
2n3KR/N1Q2rBR/2B5/2PqnB2/n1bbp1Q1/1n1k4/R1b1r1r1/1q2b2R
Q2N4/r1R3Q1/p1BrBKPp/r3Q3/3b2r1/R2n4/1qqrnk2/B2r2rB
2qr4/r2Qq1p1/1kbB4/1P1R1q2/2K1NBpn/1N1Q1pBq/2NnNb1n/8
R3B1N1/B4b1r/bN2QqB1/p6p/K1Np1kbB/2q1qb2/1b5P/2nn2Q1
b1b3Q1/1qn1k3/r4b2/1B3P2/1pqB1q1N/2B1pp2/1N2r2N/Q1KrRr1N
2Nqkr2/K1p1N3/1R3p2/1bp5/bQ2Q3/p1B5/2nPnNp1/bn1Bq1QN
2r1R2n/r1N1nr1B/3R2b1/b2B1N2/1b1nRBbr/1B1n2P1/3B2b1/K5qk
Bk3B2/6PB/Bp2BB2/p2K4/1Nn2n2/1p1nr2R/qRpp1Qr1/5nbb
5R2/1r1k3p/8/K1RP1qb1/1rBN1BB1/Br1B3N/1Bn2rnr/1bqn1r2
1RBRR2q/1n2pB1b/4r1B1/1np1P1n1/1R2b2r/B4b2/3NKb2/1q1nNnk1
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3b3b/k1r1q2n/3n2Kp/R4R2/p1rpPQ1R/2bB4/PQR3NN/2Bq1r1b
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1q2k1br/P4R2/NB1p1r1R/1KR1QB1p/5p1P/1N4p1/BRP2Pr1/n4Q1rAjedrecista.