# Questions tagged [mathematics]

Questions concerning the mathematics of the game of chess, or chess-related mathematical pursuits.

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413 views

### Puzzles that highlight the finiteness of the chessboard

I'm curious as to whether there are compositions of chess puzzles/problems where the solution really emphasizes the boundedness of the board. Put differently, cases where the solution would be ...
490 views

### Longest possible game of Horde Chess

The longest possible normal game of chess has been calculated, but I want to know what the longest possible game of Horde Chess is. What makes this an interesting case is the crazy amount of pawn ...
393 views

### What are unsolved problems in chess? [closed]

Are there any problems in chess that remain unsolved?
268 views

### How many two move checkmates exist?

I am well aware that Fool's Mate exists. But how many two move checkmates exist in total?
547 views

### Pal Benko's final problem

EDIT: Link added One of my chess heroes, Hungarian GM Pal Benko, is no longer with us. He overcame many difficulties in his early life to live one which was rich and full, and reached the age of 91 ...
298 views

### How many unique pawn structures are possible?

In an effort to classify chess games, I thought it would be useful to identify a set of positions by their unique pawn structure that would form a family of positions. There are indeed many positions ...
404 views

### Minimal number of moves before castling in Chess960

As you know, in the initial position Chess960 the king lies between the two rooks. Here are the rules for castling. For a starting position P, let n(P) be the minimal number of moves that White needs ...
2k views

### Chess480: Why 480?

From wiki: Lewis has named this chess variation "Chess480"; it follows the rules of Chess960 with the exception of the castling rules. Although a Chess480 game can start with any of 960 ...
103 views

### Relation between ratings for Chess960 and standard chess

Is there any relation or any attempt at discovering a relation between Chess960 ratings and standard chess ratings? This can be for GMs, for (any)M's, for pro players, or for all players on lichess/...
113 views

### A mobility puzzle

[FEN "8/8/8/8/pp6/kp6/1p6/1K6 w KQkq - 0 1"] This is a well-known position: both sides are stalemated (if it's their move). It's the lightest position with this property. Devise the ...
407 views

### EXTENDED New Year's Math Riddle: 5-fold repetition & 75-move rules

This puzzle extends Rewan Demontay's New Year Math Riddle. Rewan asked (essentially): if the players co-operate, what's the largest number of times a given diagram can occur in a game before reaching ...
1k views

### How many moves needed for a knight to go from any square to any other square?

I want to find a sequence of knight moves that has a given origin and destination squares, preferably with the minimum number of moves. I can iterate over knight moves and collect visited squares, ...
154 views

### Number of distinct checkmate positions

What is the number of distinct checkmate positions in each of the following categories given the following conditions? We are not interested in whether the position can be forced or not or even what ...
258 views

### How many games of optimal chess are there?

Assume that we have weakly solved chess, in the sense that we have an algorithm that when used, allows us to force a win if possible, and if that is not possible, force a draw. Define a ply as optimal ...
140 views

### How many sensible games of chess are there?

Analytic procedures for estimating the number of possible games of chess typically reason that on each ply, a player has some limited number of sensible options, O, and the average game of chess lasts ...
1k views

### Square Free Position

[FEN ""] For the purposes of this question, a “square” is a set of four occupied cells on the chessboard whose centres form a geometric square. The cell may have any orientation. For ...
196 views

### The Chess960 analogue in shogi, xiangqi, etc

Fischer said In chess so much depends on opening theory, so the champions before the last century did not know as much as I do and other players do about opening theory. So if you just brought them ...
3k views

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

The 8 queen's puzzle is a well-known puzzle involving placing 8 queens on a chessboard so that no queen attacks another queen. It is more or less obvious that 8 is the maximum number of queens that ...
252 views

### 16 roses, 16x16 board; maximum double-attacked squares while all 256 squares are attacked at least once?

The rose is a piece that is best described as a "circular nightrider", able to reach up to 32 other squares from the center of a 13x13 area by making consecutive knight jumps in a rotating fashion in ...
473 views

### Which collections of pieces are legal?

Call a collection of (white and black) chess pieces legal if it occurs in the position of a legal chess game. For example, KQRRBBNNPPPPPPPPkqrrbbnnpppppppp is the collection at the beginning of the ...
14k views

### Has anyone attempted to characterize chess mathematically?

From years of being student, researcher, and practitioner of chess, I feel like it possesses completeness and elegance that are found in mathematical proofs and theories. Thus, it seems only natural ...
1k views

### Longest Sequence For Any Piece To Reach A Certain Square

What is the longest sequence of moves (by both White and Black) wherein White attempts to move one of their pieces from one square to another? Assume Black makes every move optimal towards the purpose ...
160 views

### CNN autoencoder for feature extraction for a chess position

I'm currently building a chess engine in my free time while entering my second year of my master's studies in computer science. For a class we need to perform a litterature survey of a topic and ...
486 views

### How many possible chess moves are there?

Starting from the standard initial position, how many possible legal chess moves are there in this and all possible following positions? Say, ignoring the starting position, a bishop can move to 32 ...
226 views

### A New Year's Math Riddle

Here's a fun and clean riddle for the New Year! I found the material for it on Krabbe's site. I will provide the link when it is solved. How many times can this position be repeated, with both sides ...
227 views

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

I'm interested in short games that reach a King vs King endgame. Sam Lloyd reached King vs King after only 17 moves. This has since been improved to 16.5 moves. (See this and that.) One can show that ...
3k views

### What is the number of legal positions in a chess game?

There have been several questions regarding the possible number of chess games. for example; Database of every possible move in chess. However,can there be an estimate on the number of "allowable&...
2k views

### How to calculate the number of combinations for Chess960 positions

Can someone derive how do we get the total number of 960 possible positions for Chess960?
2k views

### How to solve this puzzle of Martin Gardner?

Martin Gardner mentioned the following "chess" problem in his column, Mathematical Games, for Scientific American (he got it from Smullyan, who learned about it from a high school math ...
14k views

### Longest chess game possible (maximum moves)

What is the longest chess game possible in terms of chess moves? I read somewhere that there is a theoretical maximum of 5949 moves. But I don't see any proof and I don't think it is correct. Can it ...
235 views

### Knights necessary to mate king on large finite board

How many knights would be necessary to mate opponent's king on large finite boards like billion times billion squares? Are they fast enough to surround opponent's king in very limited constant count ...
2k views

### Least number of moves to achieve a complete switchover of nonpawns in starting setup

[fen "RNBQKBNR/8/8/pppppppp/PPPPPPPP/8/8/rnbqkbnr w - - 0 1"] During a comp, my friend an I had a team kill and we decided to make this position(legally) before drawing. Then we continued to find ...
587 views

### Checkmating with two bishops

Let White have the two bishops. Given any setup, has the maximum number of moves to forced mate been calculated on an nxn board (for large enough n)? If not, are there any bounds on this number? Of ...
51 views

### What arrangement of 8 each of these 4 different compound leapers has the “strongest coverage properties” on a 16x16 board?

The 4 compound leapers I'd like to look at are: (1,2), (0,3), (2,2); knight+threeleaper+alfil (1,2), (0,2), (1,1); knight+kirin (1,2), (1,3); knight+camel (1,2), (2,3); knight+zebra What arrangement ...
424 views

### How to mathematically determine if king and pawn versus king endgame is a draw or a win?

I am aware of questions such as this one which discussed a method of determining if a king pawn vs king endgame is a draw or not. The accepted answer gave detailed guidelines but is not mathematically ...
3k views

### Arrange 5 non-attacking knights on a 5x5 toroidal board

The celebrated British mathematician and lover of public transport, Simon Norton, was one of those who passed in 2019. He was the subject of a wonderful biography and one of my favourite human beings: ...
1k views

### How to calculate the number of positions in chess?

For layman, can someone explain how you calculate the number of available positions in a chess game, and how do you get the huge number of 10^120 or 10^46?
143 views

### Calculating the precise number of combinations of possible moves from a given position

I was surprised Bobby Fischer couldn't find a mate in 3 within 30 minutes in this problem by Pal Benko. This made me wonder-how many actual moves are possible with so few pieces left in only 3 moves? [...
1k views

### Longest possible FEN

What is the longest possible FEN? The reason for that question is, that I am designing a PGN viewer (again), and I have noticed that the FEN is dependent on the width of the board (the FEN test area ...
473 views

### Fastest King vs King endgame

What is the fastest possible game that ends in a King vs King endgame? Please post a game and tell me the number of half-moves you've achieved. By some simple logic, I can prove that this number is ...
180 views

### What is the highest mobility score of a board without a checkmate, stalemate, or draw?

I consider the mobility score of a board to be the number of legal moves one player could make less the number of legal moves the other player could make. 8/7k/2Q2K2/8/6b1/8/8/8 w - - 0 1 Using the ...
1k views

### How does Elo rating gain scale with games played?

I only take two players starting out at 1500 Elo and have them play against each other with one player winning all games. Also I leave out any artificially introduced cut-offs. So the winner is ...
130 views

### Longest non-cooperative game of chess [closed]

Typical arguments for the longest possible game of chess take a form something like this. 1) Each player moves their knights back of forth 49 times 2) Each player moves one pawn forward one square ...
7k views

### Stalemate situation with all pieces on the board

Suppose all the pieces are on the board. Does there exist a position such that A) one of the players can't make any move (a stalemate)? B) neither player can make any move (a double stalemate)?
124 views

### Longest sequence before a certain capture

Inspired by a recent question, what is the longest possible sequence before White can make a certain capture? Basically, White is trying to capture a chosen piece, and Black optimally defends that ...
355 views

### Is the number of legal chess positions odd or even?

Is the number of legal chess positions odd or even? Two positions are not the same if they differ in castling rights (i.e. whether K or R have actually moved) or en passant capability (i.e. whether ...
490 views

### What's the maximum number of moves that can result in a threefold repetition from an arbitrary position?

Assume that the fifty-move rule does not require a player to claim a draw. Instead, the game is automatically declared a draw if no pawns have been moved and no pieces have been captured in the last ...
657 views

### Biggest Possible Material Advantage But Still Losing

Assuming: Queen=9, Rook=5, Bishop=3, Knight=3, Pawn=1 The below position is evaluated at +411 for White but he still loses by force (mate in 9, pawn promotes to a bishop just for the fun of it) Is ...