55

Here is an example of a 12-move game after which White (to move) is stalemated. All 32 units (pieces and pawns) are still on the board. The original version of this concept game was created by Charles Henry Wheeler, and published in Sunny South in 1887, according to Edward Winter's C.N. 3679. Samuel Loyd is often, and wrongly, given credit. [Title ""] [...


35

There are many different aspects of chess which can be formalized mathematically. Since the 19th century at least, chess has been mined as a resource to drive mathematical innovation. So when talking about a mathematical characterization of chess, it's not a single modeling that we are talking about, which grabs every feature, but rather a number of models, ...


21

As @chakerian's calculations show, 40 moves is the minimum. After a bit of puzzling, I found the solution. [FEN ""] 1. a4 {First, we need to get the rooks in position. They'll be hard to swap once the board is full.} a5 2. h4 h5 3. Rh3 Rh6 4. Rg3 Rf6 5. Rg6 Rf3 6. Rh6 Rh3 7. Ra3 Ra6 8. Rc3 Rb6 9. Rcc6 Rbb3 10. Ra6 Ra3 {Now, we can start with the ...


19

14 non-attacking bishops We may consider the white-square bishops and the black-square bishops separately. At most 7 bishops can be placed on white squares, namely, at most one bishop on each of the 7 white diagonals parallel to the h1-a8 diagonal. In fact, we can put bishops on the 7 white squares b1, d1, f1, h1, c8, e8, g8. The solution for black-square ...


18

Disclaimer: This solution is not reachable from the starting position, and is not reachable in a game of Chess960 (thanks Rewan!). [FEN "3bBNRN/2pPpPKQ/2P1P1PR/7P/p7/rp1p1p2/qkpPpP2/nrnbB3 w - - 0 1"] Why does the solution here not work? This is clearly not reachable from the starting position (because of the bishops stuck on the first rank), but the ...


15

Whilst acknowledging the comment by @SmallChess that this is pointless, it is also relatively straightforward to do. I analysed 2,539,871 games from a ChessBase mega database counting the number of moves for the next player to move before each move was played. I did not include the number of moves available after the final move of the game had been played. ...


13

For knights, the maximum is 32. Since knights can only attack the color opposite of the square they’re on, placing one on 32 squares of the same color is therefore optimal. [FEN "N1N1N1N1/1N1N1N1N/N1N1N1N1/1N1N1N1N/N1N1N1N1/1N1N1N1N/N1N1N1N1/1N1N1N1N w - - 0 1"] As for bishops, 14 is the highest possible. [FEN "B7/B6B/B6B/B6B/B6B/B6B/B6B/B7 w ...


13

You are asking for the diameter of the knight's graph. I suppose you only want it for the ordinary 8x8 chessboard. (See OEIS sequence A232007 for diameters of knight's graphs on square boards of size nxn, in particular confirming my answer 6 for the case n = 8. The answer is alleged to be ceiling(2n/3) for n > 4.) The answer is 6. It takes 6 moves to get ...


12

I noticed a simplification of @RosieF's solution: Label the squares with two-number labels as follows: 4,2 0,3 1,4 2,0 3,1 1,0 2,1 3,2 4,3 0,4 3,3 4,4 0,0 1,1 2,2 0,1 1,2 2,3 3,4 4,0 2,4 3,0 4,1 0,2 1,3 Then two knights may occupy two squares x,y and X,Y iff those squares' first numbers x and X are different and their second numbers y and Y are ...


12

Thanks for the fun question, and welcome to CSE! For this kind of construction problem, the king is obviously the piece to choose. Addendum 10/12/2020: The position that used to be here proved defective, so here is a new position, Unfortunately over a hundred moves had to be cut. Here is my new offering of 336 moves, with the White king going to b7, [Title &...


12

According to the FIDE Laws of Chess: 1.4 The objective of each player is to place the opponent’s king ‘under attack’ in such a way that the opponent has no legal move. 1.4.1 The player who achieves this goal is said to have ‘checkmated’ the opponent’s king and to have won the game That means that if white has a knight which is 2 normal knight moves away ...


11

Some cleaning up is required, I think: The number on the website you link to differs from the results published in Bonsdorff et al., Schach und Zahl. Unterhaltsame Schachmathematik. pp. 11–13. There they say that if the 50-move-rule is mandatory the longest possible game (i.e. where both players cooperate to achieve the weird goal of a game of maximal ...


11

This is a famous task, originally tackled by Sam Loyd and only improved a century later. See http://www.chessvariants.com/problems.dir/twokingstask.html, which gives the refinement by Ponzetto: [FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1.e4 d5 2.exd5 Qxd5 3.Bd3 Qxa2 4.Bxh7 Qxb1 5.Bxg8 Qxc2 6.Bxf7+ Kxf7 7.Rxa7 Qxc1 8.Rxb7 Rxh2 9.Rxb8 ...


10

Shouldn't the minimum be 40?. I don't have any result to show yet but it seems that: Pawns (black and white): 8 moves Rooks (black and white): 10 moves Kings (black and white): 7 moves Queens (black and white): 3 moves Knights (black and white): 8 moves Bishops (black and white): 4 moves I could be wrong Something to consider is that the e pawns for ...


10

TL;DR: a solution! [FEN "rnbq1bn1/p2pp1p1/1p2r1Nk/2p4p/6PN/1BPP1p1P/PP2PP2/R1BQK2R w - - 0 1"] 32 pieces, no squares. According to OEIS sequence A240443 and this particular pair of examples, 34 is the maximum possible number of square-free points on an 8x8 grid. As long as each file has at least two points in ranks 2-6, this is likely to give us a ...


10

Due to the geometry of pawns moves, 8 seems to be the maximum for singular pawn moves. Here is an example game for "e5." [FEN ""] 1. e4 Nc6 2. e5 Nb8 3. e6 dxe6 4. d3 e5 5. Nc3 e4 6. dxe4 Nc6 7. e5 Nb8 8. e6 fxe6 9. Nb1 e5 10. f3 e4 11. fxe4 e5 12. Nf3 Nc6 13. Nxe5 Nb8 14. Nf3 Nc6 15. e5 Qd3 16. cxd3 Bf5 17. e6 Be4 18. dxe4 Nd4 19. e5 ...


9

41 Half moves, not a real game The first possible capture is indeed on the third half-move. After that, a perfect game would be purely captures. By counting the moves which don't involve a capture, you can show how close to a perfect king v king you got. Giving check is bad, unless the opposing king can take a piece while moving out of check (unlikely, if ...


9

You're counting the same position a couple of times and not including the rule that the bishops have to be on opposite colors. (And possible that the king must be between the rooks.) The actual number is 4 x 4 x (6 x 5) / 2 x 4 4 for the LSB 4 for the DSB (6 x 5) / 2 for the two knights (halved due to account for the same type of piece being placed) 4 ...


9

Here are some starting spots in reading up on Game theory which is the mathematical tool that would be most appropriate for making claims about chess. This is a light history of early game theory. Chess is a "perfect information game" and there are some interesting things one can claim about this category of games. See this for example. I would also read ...


9

This is my answer :) Spoiler alert: Answer is in a comment in the last line of the code block using System; namespace Toroidal5Knights { class Program { static int solution = 0; static bool[,] board = new bool[5, 5]; static int[] dx = { -1, 1, 2, 2, 1, -1, -2, -2 }; static int[] dy = { -2, -2, -1, 1, 2, 2, 1, -1 }; ...


9

Label the squares with two-number labels as follows: 4,2 0,3 1,4 2,0 3,1 1,0 2,1 3,2 4,3 0,4 3,3 4,4 0,0 1,1 2,2 0,1 1,2 2,3 3,4 4,0 2,4 3,0 4,1 0,2 1,3 Then two knights may occupy two squares x,y and X,Y iff those squares' first numbers x and X are different and their second numbers y and Y are different. Regard these numbers as modulo 5. Then the two-...


9

We can easily get a reasonably good upper bound on the number of positions. At any point in time, each player has 16 pieces of which the 8 pawns can perhaps be promoted to a knight / bishop / rook / queen. Consider all captured pieces to be off the board but part of the position (this would increase the number of positions but that is fine because we only ...


9

Answer 1: Yes, this is possible. Here is an example game in 27 moves: [FEN ""] 1. a4 a5 2. b4 b5 3. bxa5 bxa4 4. Rxa4 Nc6 5. g4 Nxa5 6. Rxa5 Rxa5 7. h4 g5 8. hxg5 h5 9. gxh5 d5 10. c4 Nf6 11. cxd5 Nxd5 12. e4 Bb7 13. exd5 Bxd5 14. Nc3 e6 15. Nxd5 exd5 16. Bg2 c5 17. Bxd5 Rxh5 18. d4 Rxg5 19. dxc5 Bxc5 20. f4 Bxg1 21. Rxg1 f6 22. fxg5 fxg5 23. ...


8

The only old documents I find available online from the Deutsche Schachzeitung periodical are from Volumes 20, 21, 44, 45, 56, 57, which are available at the Internet Archive. So if you really are after Pauls' exact article for historical reasons, you might have to track down a hard copy of Volume 29 at a library. On the other hand, if you are primarily ...


8

I came up with this, which is 43 moves, but I think it could be optimized further: [fen ""] 1. h4 h5 2. Rh3 Rh6 3. Rg3 Rf6 4. Rg6 Rf3 5. Rh6 Rh3 6. Rh8 Rh1 7. a4 a5 8. Ra3 Ra6 9. Rc3 Rb6 10. Rc6 Rb3 11. Ra6 Ra3 12. Ra8 Ra1 13. e4 d5 14. Ke2 Qd7 15. Ke3 Qg4 16. Qf3 Kd7 17. Qf4 Kc6 18. Kd4 Qd1 19. Ke5 Bh3 20. g4 f5 21. Ke6 Kc5 22. Kf7 Kd4 23. Ba6 b5 24. Bc8 ...


8

[FEN "QQQQQbBk/QQQQQBQB/QQQQQRBK/QQQQQQQQ/QQQQQQQQ/QQQQQQQQ/QQQQQQQQ/QQQQQQQQ b - - 0 1"] 1... Bxg7 I'm getting +518 for this. That could be improved to +520 if we allow a black pawn on the 8th rank instead of the bishop, since this position is obviously already impossible for a number of other reasons. Edit: Found a better one. QQQQQQNk/QQQQQQNp/...


8

The bishops must be on opposite colors and the king must be between the rooks for Chess960. Each bishop can go on one of four squares, so 4x4. Then the queen can go on one of the six remaining squares, so 4x4x6. Now there are five squares for the knights, but the knights are interchangable, so there are 5x4/2 = 10 ways. So that makes 4x4x6x10 = 960. Now ...


8

The number of games is huge but finite, and estimates have been made based on a number of assumptions. But that question has been asked before already, so I won't go into the details here. A short answer given on Wikipedia is at least 10123, based on an average branching factor (moves per position) of 35 and an average game length of 80; after only 10 plies (...


8

The three first steps: There are 90 positions where n(P)=1 For White to be able to castle short at once, he needs to have Kf1, Rg1 in the initial position. Then there are 5 possible spots from the second rook, 3 for one bishop and 2 for the other one, and finally 3 remaining spots for the queen. The knights go to the last two spots. There are 64 positions ...


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