# Tag Info

### Stalemate situation with all pieces on the board

Here is an example of a 12-move game after which White (to move) is stalemated. All 32 units (pieces and pawns) are still on the board. The original version of this concept game was created by Charles ...
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### Has anyone attempted to characterize chess mathematically?

There are many different aspects of chess which can be formalized mathematically. Since the 19th century at least, chess has been mined as a resource to drive mathematical innovation. So when talking ...
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### Is there a mathematical theorem that you can use to win a chess endgame?

Unfortunately the (original) question was a bit ambiguous and arguably still is. I try to sort the different interpretations out: Chess fulfils all requirements of "standard" game theory (...
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### How many moves does it take a knight to move 1 square forward?

Depends on whether the square is diagonal (2 moves) or not (3) as shown in the diagram below:
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### Can a unique position be deduced if pieces are replaced by checkers (can see piece color but not type)

Maximum: [FEN "PppppppP/pppppppp/8/8/8/8/PPPPPPPP/pPPPPPPp w KQkq - 0 1"] The board must be right side up because the armies cannot switch sides without some capturing. Given this, all the ...
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### Least number of moves to achieve a complete switchover of nonpawns in starting setup

As @chakerian's calculations show, 40 moves is the minimum. After a bit of puzzling, I found the solution. [FEN ""] 1. a4 {First, we need to get the rooks in position. They'll be hard to ...
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### Maximum number of knights populating a chess board so that no knights are attacking one another

14 non-attacking bishops We may consider the white-square bishops and the black-square bishops separately. At most 7 bishops can be placed on white squares, namely, at most one bishop on each of the 7 ...
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### Is there a mathematical theorem that you can use to win a chess endgame?

Such a theorem doesn't exist. The closest is the strategy-stealing argument that can be used to show that for many games, the second player does not have a winning strategy. The gist of the argument ...
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### Checkmate in ω² moves with finitely many pieces

Introducing a position of game value ω² with finitely many pieces In the following position with finitely many pieces, White has mate-in-ω² (Black to move): (For convenience, I put the position into a ...

### What is the average number of legal moves per turn?

Whilst acknowledging the comment by @SmallChess that this is pointless, it is also relatively straightforward to do. I analysed 2,539,871 games from a ChessBase mega database counting the number of ...
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### Stalemate situation with all pieces on the board

Disclaimer: This solution is not reachable from the starting position, and is not reachable in a game of Chess960 (thanks Rewan!). [FEN "3bBNRN/2pPpPKQ/2P1P1PR/7P/p7/rp1p1p2/qkpPpP2/nrnbB3 w - - 0 1"...
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### How many moves does it take a knight to move 1 square forward?

Not to detract from db_max's answer with its marvellous picture, but here's another way to arrive at the answer. When a knight moves, it moves from a white square to a black square or vice versa. You ...
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### How many times can the same pawn move be made in a game?

Eight is the maximum due to the geometry of the board. Only five of one side and three of the other can converge on the same single square. Addendum: o@Red_Floyd2 on reddit found a faster way in 18.0 ...
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### When can legions of knights on an infinite chessboard checkmate a lone king?

If the king is at least three squares away from the infinite east-west line of knights then imagine the king starts running away to the north. At some point, after making n moves the king must be ...
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### How many moves needed for a knight to go from any square to any other square?

You are asking for the diameter of the knight's graph. I suppose you only want it for the ordinary 8x8 chessboard. (See OEIS sequence A232007 for diameters of knight's graphs on square boards of size ...
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### Fastest King vs King endgame

This is a famous task, originally tackled by Sam Loyd and only improved a century later. See http://www.chessvariants.com/problems.dir/twokingstask.html, which gives the refinement by Ponzetto: [FEN ...

### Maximum number of knights populating a chess board so that no knights are attacking one another

For knights, the maximum is 32. Since knights can only attack the color opposite of the square they’re on, placing one on 32 squares of the same color is therefore optimal. [FEN "N1N1N1N1/...
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### Arrange 5 non-attacking knights on a 5x5 toroidal board

I noticed a simplification of @RosieF's solution: Label the squares with two-number labels as follows: 4,2 0,3 1,4 2,0 3,1 1,0 2,1 3,2 4,3 0,4 3,3 4,4 0,0 1,1 2,2 0,1 1,2 2,3 3,4 4,0 2,4 3,0 4,1 0,...
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### Longest possible FEN

The FEN consists of six parts (see definition of FEN here): piece placement: at most 64 characters, one for each square plus 7 times "/" to delimit ranks = 71 characters active color: one character ("...
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### How to solve this puzzle of Martin Gardner?

According to the FIDE Laws of Chess: 1.4 The objective of each player is to place the opponent’s king ‘under attack’ in such a way that the opponent has no legal move. 1.4.1 The player who achieves ...
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### Is it possible to estimate my ELO without joining an organization or anything?

Puzzles are a notoriously bad estimator for ELO. (Tried myself, wild fluctuations.) You could run whole games by you through an engine (moar input => moar output) and check the accuracy of your ...
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### Longest Sequence For Any Piece To Reach A Certain Square

Thanks for the fun question, and welcome to CSE! For this kind of construction problem, the king is obviously the piece to choose. Addendum 10/12/2020: The position that used to be here proved ...
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### The number of legal positions after n plies

Such counting of moves is called perft and it is used for testing move generation of chess engines. You can find a table with your answer up to n = 15 here: https://www.chessprogramming.org/...
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### Biggest Possible Material Advantage But Still Losing

[FEN "QQQQQbBk/QQQQQBQB/QQQQQRBK/QQQQQQQQ/QQQQQQQQ/QQQQQQQQ/QQQQQQQQ/QQQQQQQQ b - - 0 1"] 1... Bxg7 I'm getting +518 for this. That could be improved to +520 if we allow a black pawn on the 8th ...
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### Biggest Possible Material Advantage But Still Losing

Just for reference: The maximum number that is actually achievable in a real game is of course when one side promotes all its pawns to queens and gets mated by a single pawn. The value would then be 9*...
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### Approximating a rating for too strong engines

There're a few ways to do it: Add 400 rating to the top opponent and take that as an approximation (used by USCF) Add a draw for the engine against itself Assign an infinite rating Use linear ...
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### Least number of moves to achieve a complete switchover of nonpawns in starting setup

Shouldn't the minimum be 40?. I don't have any result to show yet but it seems that: Pawns (black and white): 8 moves Rooks (black and white): 10 moves Kings (black and white): 7 moves Queens (...
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### How to calculate the number of combinations for Chess960 positions

You're counting the same position a couple of times and not including the rule that the bishops have to be on opposite colors. (And possible that the king must be between the rooks.) The actual ...
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