# Questions tagged [mathematics]

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

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

Are there any problems in chess that remain unsolved?
2k 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 ...
522 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 ...
460 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 ...
303 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 a billion times a billion squares? Are they fast enough to surround the opponent's king in a constant number of ...
609 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 ...
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?
820 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 ...
537 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 ...
7k 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 ...
1k 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?
494 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 ...
672 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 ...
184 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 ...
122 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?
87 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 ...
311 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 ...
612 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 ...
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 ...
777 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 ...
280 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 ...
140 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?
908 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).
519 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 ...
290 views

### Is there a formula that converts a number from 0 to 959 to a corresponding chess960 position?

I am looking into a website with starting positions for chess960. 000 BBQNNRKR 001 BQNBNRKR 002 BQNNRBKR 003 BQNNRKRB 004 QBBNNRKR 005 QNBBNRKR ... 959 RKRNNQBB So there is a table that converts any ...
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&...
615 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 ...
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. ...
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 ...