60
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 ...
- 6,265
39
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 (...
- 15.4k
38
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 ...
- 11k
30
votes
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:
- 787
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 ...
- 24.8k
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 ...
- 3,762
20
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 ...
- 26.1k
18
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 ...
- 1,761
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
18
votes
Accepted
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 ...
- 473
16
votes
Accepted
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 ...
- 17.1k
16
votes
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 ...
- 6,265
15
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 ...
- 3,762
14
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 ...
- 751
14
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/...
- 17.1k
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 ...
- 94.5k
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 ...
- 15.4k
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,...
- 349
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 ...
- 17.1k
12
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 ("...
- 20.7k
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/...
- 164
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 ...
- 7,504
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 ...
- 26.1k
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 (...
- 790
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*...
- 20.7k
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 ...
- 3,112
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 ...
- 10.5k
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 ...
- 780
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,...
- 11k
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, ...
- 2,081
Only top scored, non community-wiki answers of a minimum length are eligible