Questions tagged [mathematics]

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

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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 ...
Yash Jain's user avatar
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11 votes
2 answers
3k views

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-...
paul garrett's user avatar
2 votes
1 answer
176 views

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 ...
Flowy Poosh's user avatar
1 vote
1 answer
104 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 ...
anti - Marshall's user avatar
3 votes
1 answer
158 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 ...
Hauke Reddmann's user avatar
1 vote
1 answer
100 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?
Prashant Akerkar's user avatar
3 votes
1 answer
127 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, ...
Hauke Reddmann's user avatar
2 votes
0 answers
47 views

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 ...
Laska's user avatar
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4 votes
1 answer
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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 ...
Hauke Reddmann's user avatar
14 votes
1 answer
3k views

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 ...
Andreas Tsevas's user avatar
3 votes
1 answer
246 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 ...
Andreas Tsevas's user avatar
-1 votes
1 answer
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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 ...
Steve Bennett's user avatar
2 votes
0 answers
93 views

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 ...
Rewan Demontay's user avatar
2 votes
1 answer
63 views

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 ...
Rewan Demontay's user avatar
6 votes
4 answers
630 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 ...
Steve Bennett's user avatar
6 votes
3 answers
245 views

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 ...
Steve Bennett's user avatar
1 vote
1 answer
98 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 ...
Hauke Reddmann's user avatar
2 votes
1 answer
108 views

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 ...
Fede Rico's user avatar
-1 votes
1 answer
113 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 ...
BCLC's user avatar
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1 vote
2 answers
264 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 ...
BCLC's user avatar
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0 votes
1 answer
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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 ...
BCLC's user avatar
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1 vote
2 answers
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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 ...
BCLC's user avatar
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7 votes
3 answers
4k views

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?
danday74's user avatar
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2 votes
2 answers
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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 ...
Hauke Reddmann's user avatar
11 votes
1 answer
435 views

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 ...
Marco Ripà's user avatar
5 votes
2 answers
188 views

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 ...
Benjamin Wang's user avatar
12 votes
3 answers
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:...
stevec's user avatar
  • 2,087
-1 votes
2 answers
358 views

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 ...
spsy's user avatar
  • 107
1 vote
2 answers
1k views

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.
user2925716's user avatar
10 votes
4 answers
3k views

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'...
M1976's user avatar
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2 votes
2 answers
256 views

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: .------------------------------------------------------ | ^ | ...
Carlo Wood's user avatar
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 ...
DKozl's user avatar
  • 221
-1 votes
1 answer
465 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 ...
BCLC's user avatar
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10 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?
user avatar
4 votes
1 answer
390 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 ...
BCLC's user avatar
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2 votes
1 answer
737 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 ...
BCLC's user avatar
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4 votes
1 answer
365 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 ...
Hauke Reddmann's user avatar
2 votes
1 answer
154 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 exd4 3.e5, there is one unoriginal pawn move, ...
Steve Bennett's user avatar
1 vote
1 answer
163 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 ...
Autistic's user avatar
  • 399
14 votes
2 answers
741 views

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 ...
Laska's user avatar
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5 votes
2 answers
3k 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....
Steve Bennett's user avatar
1 vote
1 answer
225 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 ...
David Gladson's user avatar
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 ...
BCLC's user avatar
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4 votes
1 answer
158 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 ...
Hauke Reddmann's user avatar
6 votes
2 answers
5k 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, ...
eguneys's user avatar
  • 473
4 votes
1 answer
405 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 ...
Playingwithnumbers's user avatar
4 votes
2 answers
311 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 ...
Nathan's user avatar
  • 283
1 vote
1 answer
361 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 ...
Nathan's user avatar
  • 283
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 ...
TSLF's user avatar
  • 325
7 votes
1 answer
920 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/...
BCLC's user avatar
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