Questions tagged [mathematics]

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

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8
votes
1answer
253 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 ...
3
votes
1answer
95 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 ...
6
votes
3answers
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 ...
12
votes
6answers
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: ...
23
votes
5answers
13k 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 ...
4
votes
1answer
226 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 ...
28
votes
4answers
5k 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)?
5
votes
0answers
111 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 ...
5
votes
2answers
232 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 ...
0
votes
2answers
107 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 ...
2
votes
1answer
72 views

Knight promotion percentages

Knight promotion is the second most common promotion. Most of the time it is to give an otherwise ungivable check. But it doesn’t always come with check. Statiscally speaking, what percentage of ...
2
votes
1answer
254 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 ...
4
votes
1answer
126 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 ?
2
votes
0answers
63 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 ...
3
votes
3answers
530 views

Of all the possible legal chess games, how many end in white win, draw and black win (assuming 50 move rule)?

If both sides make moves completely randomly but legally, how many different games are there? Since the 50 move rule applies, this number should be finite (but HUGE). For example, there are 20 ...
3
votes
2answers
145 views

How many different knight's tours are there?

How many solutions does the Knight’s Tour puzzle have when the knight first moves from square a1, through all squares (on each square once), and return back to a1? Please give some solutions. Also, ...
-4
votes
1answer
134 views

What is the total number of KNN vs K checkmate positions?

What is the total number of KNN vs. K checkmate positions? Assume that White is checkmating. Firstly the Black king must be mated on a corner. It is impossible to do it on the edge. The calculations ...
3
votes
2answers
303 views

Piece And Pawn Only Massacre Games

Just another theory thought of mine. I believe I have found the fastest way for each variation. Am I right or am I wrong? EDIT: Now a sub-category of Piece Massacre is open in which pawn captures are ...
3
votes
3answers
252 views

What is one quick way to reach a mutual, symmetrical stalemate position?

I’m just wondering what one quick way, not the fastest, is reach a position where both sides have worked together to create a mutual stalemate here neither side could move on their turn, and the ...
1
vote
2answers
207 views

How many possible two-move checkmates are there?

Yes, I am well aware that Fool's Mate exists. But how many other possible two move checkmates exist in chess?
8
votes
3answers
632 views

What is the average number of legal moves per turn?

For example in the first turn there are 20 possible legal moves (16 pawn moves and 4 knights moves). This number increases in the middlegame and then decreases in the Endgame. Are there some ...
6
votes
3answers
400 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 ...
4
votes
3answers
360 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 ...
2
votes
3answers
305 views

FEN inverter website/page?

This board shows the black pieces on the bottom, but the white pieces are supposed to be there; I accidentally set up the board so that white's pieces are on the black side. I am looking to flip the ...
5
votes
2answers
375 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 ...
13
votes
3answers
393 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 ...
4
votes
2answers
5k views

How do you calculate your Tournament Performance Rating?

How do you calculate your Tournament Performance Rating? I can think about two different ways to answer my question: Do you know the formula for calculating your TPR (FIDE or USCF)? Can you link me ...
3
votes
1answer
967 views

What is the minimum number of pieces needed in a chess set to account for all possible promoted pieces?

How many pieces would be needed in a chess set where all possibilities for promoted pieces are included if pawns are no longer used after they promote?
2
votes
1answer
326 views

Estimating probability of player playing move in the opening

For position you never played, book probability is perfectly fine for me. The question starts when you already played position x-times. I'm looking for function that will update book probabilities ...
6
votes
6answers
511 views

How to determine if a position is open or closed in a “rigorous” fashion?

I am creating a set of chess positions to be used [as a data set] for a project. It's based on supervised machine learning, but I need to have labeled data (see more italicized text below) before any ...
-1
votes
2answers
172 views

False calculation of chess moves

At the start I must admit that I am not the brightest chess player, so this just might be a stupid question but: Calculated Chess moves have been determined to be infinite, as I am given to ...
5
votes
2answers
112 views

Endgames: How was “maximum moves required” determined before tablebases?

To use a simple example, tablebases show that KQ vs. K is won in at most ten moves. But chess books going back many decades, well before tablebases, say the same thing. How was this calculated?
-2
votes
2answers
86 views

How to calculate which opening move is modern

What approach would you recommend to calculate current tendency in opening position? You have set of (move+year) values from games. Amount of total games is unknown but will be somewhere between 1 ...
7
votes
1answer
229 views

Mathematical formula to get how broad opening repertoire is

programmers and mathematicians! I'm looking for some opinions how to build meaningfull formula that represents broadness of opponent's repertoire. At the moment I do it like this: I filter all ...
6
votes
1answer
391 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? If not, are there any bounds on this number? Of course, information on extensions of ...
2
votes
1answer
429 views

What's the average piece density?

In a position, the piece density is equal to the total number of pieces on the board (Whites pieces + Black pieces) divided by the total number of squares (64). At the beginning of every game, the ...
3
votes
1answer
210 views

Taxicab distance for bishops

I have two questions related to defining distances in chess. On this wiki page, this subject is briefly touched, by saying that the taxicab (also called Manhattan) distance is used for rooks, which as ...
0
votes
0answers
126 views

What is the maximum number of possible moves a player can choose from? [duplicate]

Let's say you arrange a position that gives a player the maximum number of choices among possible moves. How does that position look like and how many possible moves are in that position?
3
votes
1answer
812 views

How many positions are there in n-pieces endgame tablebases for different values of n? [duplicate]

Note that I don't want the exact values, I only need approximations. And I am only interested in small values of n (not more than 12).
10
votes
3answers
457 views

Where can I find Pauls' 1874 article on the n-queens problem?

Where can I find the following article, which concerns the n-queens problem? E. Pauls, Das Maximalproblem der Damen auf dem Schachbrete, II, Deutsche Schachzeitung. Organ f¨ur das Gesammte ...
8
votes
4answers
576 views

Does playing tournament chess help one in the field of chess-related math?

A question of personal interest. Is "visualization" of various types of boards and graphs helped along by competitive, tournament play? Of course, I'm assuming play beyond simply knowing how the ...
4
votes
3answers
2k views

How many possible combinations for the number of pieces on the board?

My question concerns the total material available on the chess board (black+white pieces). At the very beginning of a game, there are 2 Kings, 2 Queens, 4 Rooks, 4 Bishops, 4 Knights and 16 pawns. ...
4
votes
4answers
1k views

Longest path for knight

Suppose the knight is on square X and he wants to get to square Y, and in order to do that he has to move N moves. how big could N be? Could that be solved using a mathematical equation? Say Nf3 ...
18
votes
1answer
2k views

How many such positions are there? (chess/math checkmate puzzle)

I'm interested in this kind of position: There are only 4 pieces on the board. If Whites goes first, they can checkmate in one move. If Blacks goes first, they can checkmate in one move. For example: ...