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## Hot answers tagged mathematics

53

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 ""] [...

31

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 knights. ...

17

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 ...

14

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. ...

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 ...

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 ...

11

Thanks for the fun question, and welcome to CSE! For this kind of construction problem, the king is obviously the piece to choose. Here is my offering of 458 moves. White wants to move their king to the square a6, but it will take a long time. I used a bit of a cheat, given how the replayer works, to show the double step for White’s a1 pawn. [FEN "...

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 ...

9

Your second basic position allows for 4 more variants beyond those you already gave, indicated by the following diagram: [FEN "1q6/1q6/1q6/1q6/Q7/K7/8/1k6 w - - 0 1"] That brings the tally for "basic positions" to 25. Whether that addition makes the list exhaustive or not I'm not completely sure (though I think it does). In any case, whatever the number ...

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

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 ...

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

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

TPR calculators and expected rating change. http://englishchess.org.uk/Juniors/tournament-performance-calculator/ http://www.uschess.org/content/view/13146/836/ Basically Performance rating is the average of your opponent's ratings with an adjustment based on the score of the game. For each win, you add your opponent's rating + 400, a draw is just your ...

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

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 ...

7

Let's count. Starting set 2 rooks 2 bishops 2 knights 1 queen 1 king (8 pawns) Max promotions 8 rooks 8 bishops 8 knights 8 queens Total In total you would need 40 pieces and 8 pawns for each color, so 96 units for black and white together. Edited to bullet format to "look" better.

7

What you are looking for is called the branching factor, and I've always seen the number 35 mentioned, but I don't know what the original source is. I guess someone estimated it some 50 years ago by counting the number of moves in a number of random positions from games, and then it became "common knowledge". The number 35 is reasonable enough in practice, ...

7

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 (...

7

It would seem that the longest games would end up being the most numerous, by far. On any particular move, if you end a game, that's one game, but if you let it continue, it's many games. It would seem, therefore, that the greatest number of games would continue until one side had a single piece left. The remaining piece should not be a knight or bishop, as ...

7

To answer the second part of your question: Suppose all figures are on the board. Does there exist a transposition of figures such that both of the opponents can't do any move (a stalemate)? No, this is not possible. Pieces are simply too mobile for this, so for a stalemate you need to hem them in (like the white queen in @RosieF's answer) or pin them on ...

7

The question has two parts to it, and Rosie F perfectly answers the first part. Regarding only the second part, the question asks for any possible position where all 32 pieces are stalemated, disregarding legality of such positions. @im_so_meta_even_this_acronym's wonderful answer proves that it indeed possible. However, I want to focus specifically on only ...

7

There are essentially six solutions that are rotationally and translationally unique. Two of them yield five translations and five more rotated translations each, for a total of twenty: NNNNN ..... ..... ..... ..... N.... .N... ..N.. ...N. ....N The remaining four unique solutions can be translated to each of the 25 positions on the board, but are ...

7

(EDIT: substantially revised because I hadn't thought carefully enough about Rosie F's labeling. The result is even cooler!) Thanks for all the great answers. I would like to add my own solution. It's key to label squares in the torus, as Rosie F and Brilliand did: 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,...

6

The furthest distance is from one corner to the opposite one. You can get from a1 to h8 in six moves, for example a1,c2,d4,f3,e5,g6,h8. There are few pairs of squares that take more than four moves.

6

To the extent that "chess-related math" requires any expertise in chess beyond the rules(*), tournament play is probably not as useful as solving and composing chess problems/studies. Typical positions in tournament play are too complicated to evaluate with mathematical certainty, while problems and studies should and usually do have rigorous proofs of ...

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