Questions tagged [mathematics]
Questions concerning the mathematics of the game of chess, or chess-related mathematical pursuits.
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Ecological footprint of chess
Chess must be a very "green" sport, the resource demand is minimal.
But then, lately online chess is in vogue, and servers eat amps.
Any estimation how much energy online chess is consuming ...
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Checkmate in ω² moves with finitely many pieces
The post Checkmate in ω
moves? has inspired a lot of discussion and mathematical research into positions on an infinite chessboard where white can mate in a transfinite amount of moves. The current ...
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Mate-in-N problems where N scales indefinitely with board size
Motivation - Long range mating patterns
This question is inspired by my recent interest in infinite chess and the post on MathOverflow Checkmate in ω moves?. Since my chess endgame skills are quite ...
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Longest sequence of moves where White's number of available moves does not change
There are challenges for the longest sequence of moves in which white's available moves increase or decrease. This challenge is for the longest sequence of moves (from the opening position) in which ...
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What is the longest sequence of increasing White moves with only White pieces?
Inspired by a comment by @Evargalo on this question, I ponder how long a sequence can last where each turn, White has more available moves than the previous turn, in a situation with only White pieces ...
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Is there a Chess 960 position in which White's moves can be decreased to zero without check?
Inspired by this question, koedem proved that, from the starting position, White's legal move amount can be decreased sequentially from 20 to 0 in 20 moves. However, doing so required some checks. I ...
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Longest sequence of moves where white's number of available moves decreases
An easier version of this
White has 20 available moves at the start of the game (from the standard position). After 1.h3 h5, white has 19 available moves. (The h pawn has lost the ability to move two ...
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Longest sequence of moves where number of available moves increases
White has 20 available moves at the start of the game. After 1.a4 e5, white has 21 available moves. (The rook has gained 2 moves, while the pawn has lost 1).
What is the longest sequence of moves ...
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Roundtrips with no shortcuts
In the picture, you see the longest roundtrip of a rook such that there are no "shortcuts" - imagine the target squares are given, then at each square you have exactly two other given ...
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Can you suggest book references on chess theory?
as stated in the title I am looking for structured textbooks on chess. I have followed chess.com lessons and I have a rough idea of tactics (forks, pins, skewers, sacrifices, etc) but I feel like I am ...
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Non-linearity in Armageddon bidding: Ostensibly, you give up more time in higher time control
Note: In the ff I assume everyone bids for black, not white.
In blitz (below 10min) :
I see often there's a 5min vs 4min predetermined time. No idea if this is fair, but let's pretend arguendo players ...
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Is armageddon more fair in a Vickrey auction? Eg Magnus & Hikaru bid 8:58 vs 8:59 in 2023 Airthings Masters
I believe armageddon is fair only if they bid, but just because they bid doesn't mean it's fair, i.e. bidding is necessary but not sufficient to be fair.
Which side has the advantage in an Armageddon ...
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Is Armageddon always without increment?
Robert Hess said tells Danya that no increment influences why Robert prefers white to black in Armageddon chess960. Of course it really depends on the bid, but I guess Robert would want a lower bid ...
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Is armageddon more fair if bids aren't 1-time only?
As pointed out by GeneM here
Armageddon Chess is a fair tie-break system only if the two players bid for how much time Black should have.
Question: But is it actually more fair if the bids are the ...
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How many moves does it take a knight to move 1 square forward?
How many moves does it take a knight to move 1 square forward?
For example, if I wanted to move a knight from e5 to e4 what is the minimum number of moves it would take?
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Definition of opposition
This just came up in my youth chess training.
A very general definition of "having the opposition", including even rare cases, is
A: All corners of the (axis-parallel) rectangle spanned by ...
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Open problems in chess involving maximum material imbalances
I am posting the present question since here Very unbalanced Chess Positions I have introduced three open problems (i.e., the third one, the fourth one, and the sixth one from the above), regarding ...
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How many more squares can you attack?
(Inspired by this cool problem)
Let’s say it’s White to move. Let A = the number of squares currently attacked by White, and B = the number of squares attacked by White after White’s next move.
What ...
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How often do Super GMs achieve 100% engine correlation?
There has been speculation that anomalous values of a correlation metric1 may hint to the use of outside assistance.
Example
Examination of Hans Niemann vs Matthieu Cornette surprises:
Hikaru Nakamura:...
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Do chess have deterministic win strategy like tic tac toe? I.e. would God always win knowing the whole tree of combinations? [duplicate]
Chess is a totally deterministic game.
Although we do not have computing power to pre-compute all the possible moves, all possible outcomes are determined.
So assuming God had unlimited computing ...
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The number of legal positions after n plies
What is the number of legal positions after n plies for n=1 through 5? I know that for n=1 it is 20.
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Is it possible to estimate my ELO without joining an organization or anything?
I would like to know if it's possible to get a rough idea of what my ELO is and how accurate those methods are.
For example, I did one of those "chess tests" and the results it gave me didn'...
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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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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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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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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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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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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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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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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 exd4 3.e5, there is one unoriginal pawn move, ...
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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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If repetition loses, who wins KvK?
Suppose we're playing a version of chess where you lose if you repeat once an earlier position (position = arrangement of pieces + who's move it is). We're down to the wire, and we've just arrived at ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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