58 votes

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 ...
  • 5,820
37 votes
Accepted

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 (...
35 votes
Accepted

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 ...
  • 9,679
21 votes
Accepted

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 ...
  • 24.1k
20 votes
Accepted

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 ...
  • 3,256
19 votes

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 ...
  • 22.6k
18 votes
Accepted

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"...
  • 295
16 votes

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 ...
  • 1,646
14 votes
Accepted

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 ...
  • 3,256
13 votes
Accepted

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 ...
13 votes

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/...
13 votes
Accepted

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 ...
  • 81k
13 votes

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 ...
12 votes

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,...
  • 339
12 votes
Accepted

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 ...
12 votes
Accepted

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/...
11 votes
Accepted

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 ...
  • 7,458
11 votes
Accepted

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 ("...
  • 20.5k
11 votes

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 ...
  • 22.6k
10 votes

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 (...
10 votes

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*...
  • 20.5k
10 votes

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 ...
  • 3,082
10 votes

If both sides move at random, but legally, then how many games will be a win for White or Black, if not a draw?

The number of games is huge but finite, and estimates have been made based on a number of assumptions. But that question has been asked before already, so I won't go into the details here. A short ...
  • 10.3k
10 votes

Is it possible to reach King vs King by moving each piece at most twice?

Answer 1: Yes, this is possible. Here is an example game in 27 moves: [FEN ""] 1. a4 a5 2. b4 b5 3. bxa5 bxa4 4. Rxa4 Nc6 5. g4 Nxa5 6. Rxa5 Rxa5 7. h4 g5 8. hxg5 h5 9. gxh5 d5 10. c4 Nf6 ...
  • 706
10 votes
Accepted

Square Free Position

TL;DR: a solution! [FEN "rnbq1bn1/p2pp1p1/1p2r1Nk/2p4p/6PN/1BPP1p1P/PP2PP2/R1BQK2R w - - 0 1"] 32 pieces, no squares. According to OEIS sequence A240443 and this particular pair of examples,...
  • 9,679
10 votes
Accepted

How many times can the same pawn move be made in a game?

Due to the geometry of pawns moves, 8 seems to be the maximum for singular pawn moves. Here is an example game for "e5." [FEN ""] 1. e4 Nc6 2. e5 Nb8 3. e6 dxe6 4. d3 e5 5. Nc3 e4 ...
9 votes

Fastest King vs King endgame

41 Half moves, not a real game The first possible capture is indeed on the third half-move. After that, a perfect game would be purely captures. By counting the moves which don't involve a capture, ...
  • 2,061
9 votes
Accepted

Minimal number of moves before castling in Chess960

Edit: This answer only deals with kingside castling; the figures have to be revised if you also consider queenside castling. The three first steps: There are 90 positions where n(P)=1 For White to be ...
  • 13.1k
9 votes

Stalemate situation with all pieces on the board

@Rosie F has answered the first part and @im_so_meta_even_this_acronym's answer proves that it is indeed possible with an illegal position. However, I want to focus specifically on legal positions. ...

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