# Questions tagged [mathematics]

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

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### Maximum saved moves to win caused by zugzwang

For a zugzwang where x>y and: If white has the move, White can force checkmate in x full moves If Black has the move, White can force checkmate in y full moves The score of such a zugzwang is x-y....
801 views

### How many different kinds of fairy units?

A recent question Is there a Fairy Chess Piece that Combines the Gnurider and Queen? suggested the following: Assuming we are only operating on a 8x8 board, how many functionally distinct fairy pieces ...
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### Can a unique position be deduced if pieces are replaced by checkers (can see piece color but not type)

Inspired by a video where top players guess which famous game a position is from, except pieces in the position are replaced by checker pieces (so they can see the piece color but not type). Of course ...
2k views

### Longest checkmates in (almost) fully covered boards?

What are the longest checkmates from initial positions which are (almost) fully covered in pieces? Down below I'll give examples of what I mean. An almost fully covered position can't be legal, ...
146 views

### In how many ways can a black king get checkmated by 2 rooks?

Given an nxn chessboard, how many times can a black king get checkmated by 2 rooks (occassionally with the help of a white king)? Is there a mathematical formula that can be brought up? How many ...
1 vote
161 views

### Are all moves or types of moves (see details) in chess possible?

Inspired by this video The rarest move in chess, in which moves are categorized based on the format of their algebraic notation, so for example the rarest type of move (according to the author and ...
59 views

### Up to which number of moves has the result of chess been calculated?

Chess hasn't been completely solved yet, even though it is completely deterministic. However, can't we "solve" it up to a certain number of moves (e.g., up to 50 moves)? Then, considering ...
62 views

### Incorrect upper bound on number of possible chess positions (10^41)?

I've written a script implementing a basic idea to count the number of chess positions. I've not even accounted for legality of the positions or difference in light and dark square bishops. Despite ...
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### When can legions of knights on an infinite chessboard checkmate a lone king?

On an infinite (perhaps only in one direction?) chessboard, what kind of configuration of (possibly a very large number of) knights could checkmate a lone king? I'm really thinking of an infinite-in-...
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### Longest checkmates with the biggest material disadvantages in "open" positions?

We can ask a question: What are the longest checkmates with the biggest material disadvantages? What, approximately, is the function between length of the checkmate and possible size of the material ...
1 vote
112 views

### More accurate estimate of chess games possible

I have heard of Shannon's number (10^120) which is the supposed number of possible chess games computed in the 1950s. But Shannon's estimate is criticised for including illegal moves. What is a truer ...
184 views

### How often can the initial position be repeated under FIDE tournament rules?

Recently a game was played where two GMs swapped kingside and queenside knights and then agreed a draw (and got their just dessert): How often can the starting position be repeated in a game without ...
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1 vote
105 views

### Stalemate positions on Chess board

How many stalemate positions in numbers without duplicates are possible on chess board? Is it a infinite or finite number in terms of stalemate positions count?
133 views

### "Every" move in long notation

A move here shall be defined as the FIDE super-long notation of a legal chess move: Sg1-f3, O-O, e5xd6 ep, Da8xg8+, Rd1-d6# and whatnot. I.e. starting square, ending square, capture sign, check sign, ...
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### Estimate the max length of chess game under WFCC?

The World Federation of Chess Composition (WFCC) has a Codex that specifies how the FIDE Laws apply to chess problems. Can you give a ballpark estimate of the length of the longest game possible under ...
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238 views

### 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 ...
276 views

### 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 ...
75 views

### 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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637 views

### 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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1 vote
100 views

### 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 ...
116 views

### 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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1 vote
276 views

### Is armageddon more fair in a Vickrey auction?

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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1 vote
143 views

### 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 ...
1k views

### 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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1 vote
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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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748 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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394 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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### 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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1 vote
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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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