Questions tagged [mathematics]

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

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Fastest algorithm to give mate with K+R vs K on infinite large board as function of starting position

Consider a board with only two edges that are perpendicular. Something like: .------------------------------------------------------ | ^ | ...
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11 votes
5 answers
2k views

Approximating a rating for too strong engines

Suppose we have an extremely strong engine that happens to win every game it participates in even against the most powerful engines and players. If it continues to win every game, its rating (for ...
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-2 votes
1 answer
181 views

What is white's increased advantage in chess90 as compared to chess870? (Chess960 can be split into 2 subsets, chess90 and chess870)

Here, it is shown 90 of 960 positions involve having to move a rook on 1 side in order to castle on the other side (unless the rook is somehow captured or something). These 90 positions come from 18 ...
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9 votes
3 answers
7k views

Is there a mathematical theorem that you can use to win a chess endgame?

Have mathematicians created any theorem that determines the outcome of a endgame? If so, how could it be used to win a specific endgame?
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4 votes
1 answer
260 views

Rating system (glicko, elo, etc): is 'predict the outcome of games' the same thing as 'reduce the number of upsets'?

From Lichess page 'Chess rating systems': Question: Which rating system is best? Answer: The purpose of rating systems is to predict the outcome of games. Therefore, they can be objectively better ...
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1 vote
1 answer
425 views

How many Chess960 positions exist in which castling on one side does not require moving the rook on the other side?

Update: It's chess870 (or chess869). See answer. Context/Motivation/Goal: In regular chess, you never have to move a rook (on 1 side) to castle (on the other side). My aim is to see how much of 9LX ...
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3 votes
0 answers
111 views

Number of games ever played in history?

Obviously, an exact answer can't be given, but we can't count all electrons in the universe either and physicists still gave an answer probably correct by an order of magnitude. Let's specify: Only ...
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1 vote
1 answer
68 views

How many unoriginal pawn moves can be made in a single game?

Consider an "unoriginal pawn move" to be a pawn move whose notation matches a move already played in the game. For instance, after 1.e4 e5 2.d4 exd5 3.e5, there is one unoriginal pawn move, ...
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1 vote
1 answer
84 views

Knight Moves Around the Board [duplicate]

It is well known that a lone knight can move all around the board covering each square once and only once. Is this possible when the lone knight starts only on certain squares ? How many unique ...
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12 votes
1 answer
514 views

If repetition loses, who wins KvK?

Suppose we're playing a version of chess where you lose if you repeat an earlier position (position = arrangement of pieces + who's move it is). We're down to the wire, and we've just arrived at the ...
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4 votes
2 answers
2k views

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

How many times in a game can "the same pawn move" be made? By "the same pawn move", I mean, a move made by a pawn that would be notated the same way. For instance, after 1.e4 e5 2....
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1 vote
1 answer
149 views

How to avoid the same opening move in an engine built using Python chess

I created a chess engine using the Python chess library. It has an evaluation function (piece values, piece square values, and capture values) and minimax. Alpha-beta pruning is also implemented. At ...
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6 votes
2 answers
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 ...
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4 votes
1 answer
137 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 ...
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6 votes
2 answers
2k 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, ...
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  • 473
3 votes
1 answer
244 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 ...
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4 votes
2 answers
282 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 ...
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1 vote
1 answer
187 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 ...
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7 votes
3 answers
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 ...
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6 votes
1 answer
484 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/...
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1 vote
1 answer
828 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 ...
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6 votes
1 answer
285 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 ...
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5 votes
1 answer
243 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 ...
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8 votes
1 answer
511 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 ...
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  • 596
3 votes
2 answers
720 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 ...
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  • 586
11 votes
3 answers
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 ...
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  • 221
10 votes
2 answers
4k 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 ...
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3 votes
1 answer
247 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 ...
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5 votes
2 answers
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?
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3 votes
3 answers
384 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 ...
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1 vote
1 answer
220 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? [...
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  • 1,751
8 votes
2 answers
2k 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 ...
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  • 458
1 vote
1 answer
57 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 ...
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5 votes
3 answers
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 ...
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  • 20.3k
12 votes
1 answer
520 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 ...
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4 votes
1 answer
131 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 ...
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7 votes
3 answers
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 ...
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2 votes
1 answer
222 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 ...
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  • 123
13 votes
7 answers
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: ...
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27 votes
8 answers
15k 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 ...
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5 votes
1 answer
568 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 ...
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28 votes
4 answers
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)?
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7 votes
1 answer
268 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 ...
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5 votes
2 answers
339 views

What is the maximum number of squares 8 queens and 8 nightriders can attack on a 16x16 board?

Including but still obstructed by occupied squares, what is the maximum number of squares on a 16x16 board that 8 queens and 8 nightriders can collectively attack at least once each? Even if attacking ...
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0 votes
2 answers
133 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 ...
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  • 109
4 votes
1 answer
559 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 ...
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9 votes
2 answers
632 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 ...
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2 votes
2 answers
501 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 ...
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  • 7,705
5 votes
1 answer
209 views

Leela Chess PUCT Mechanism

How do we know w_i which is not possible to calculate using the tree search only? From the lc0 slide, w_i is equal to summation of subtree of V ? How is this equivalent to winning ?
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  • 161
2 votes
0 answers
78 views

Probability of winning based on Elo rating [duplicate]

I already looked on several websites but I have not been able to get a clear and thorough answer to this question: If player A has FIDE rating x and player B has FIDE rating y, how many times, on ...
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