51

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 ""] [...


28

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, ...


16

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 ...


13

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. ...


11

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

Thanks for the fun question, and welcome to CSE! For starters, the king is obviously the piece to choose. I will answer this question assuming that the normal rules of chess are used and no special conditions are used other than what the question has. The OP has clarified that both sides need a king and first/last rank pawns can do a double step. ...


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

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 }; ...


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

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

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 ...


8

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 ...


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 ...


8

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-...


7

Now that's a refreshingly different kind of question :) It's probably easier to compose such a problem than to search for one, but that said, there's one that comes to mind: [Title "White to play and win"] [fen "8/5N1k/8/7b/1b5N/1K6/1B6/8 w - - 0 2"] Hint: Solution:


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

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,...


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 ...


6

For people stumbling on this question: volume 29 is available online though Google Books since februari 2015.


6

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, ...


6

In the pawns only, you don't need to move so many pawns. Getting the major pieces involved early seems to be fastest: [FEN ""] 1.e4 c5 2.Qh5 Qa5 3.Qxh7 Qxa2 4.Qxg7 Qxb2 5.Rxa7 Rxh2 6.Rxb7 Rxg2 7.Rxd7 Rxf2 8.Rxe7+ Kd8 9.Qxf7 Rxd2 10.Qf5 Qxc2 11.Qxc5 Qxe4+ * I was unable to find a faster version of the piece only.


5

As a chess player who currently studies computer science, I found this question very interesting so here's my thoughts on the topic. You already know how chess books categorize open and closed positions and you may also know that a semi-closed position is one with some of the characteristics of an open but not all. As you understand this makes the line ...


5

Nothing was calculated. They arrived at their results by trial and error. Some guy would start with a seemingly disadvantageous position and tried to find the shortest mate. His result remained valid until the next guy found a more disadvantageous position or a quicker mate. A long time later, engines became powerful enough to calculate simple endgames ...


5

First of all, I think this is a bad question (similar to StackOverflow "Give me the code" questions). That being said, the subject itself seemed interesting, so I threw a quick and dirty (and inefficient - feel free to optimise) script together to do this. I ran it on PGN file of world championship games between 1895 and 1985 from this page (this should fit ...


5

This depends a great deal on how (and what) you count. In a discussion at the CCRL board, Kirill Kryukov offers the following counts on the number of unique legal positions (NULP) for some small numbers of pieces: Pieces NULP 2 462 3 368,079 4 125,246,598 5 25,912,594,054 6 3,787,154,440,416 ...


5

I agree with the previously-given answer from Matthew Liu (please give it the checkmark), but I’ll add the standard notation of the solutions since you ask: 1.f3 e5 2.g4 Qh4# 1.f3 e6 2.g4 Qh4# 1.f4 e5 2.g4 Qh4# 1.f4 e6 2.g4 Qh4# 1.g4 e5 2.f3 Qh4# 1.g4 e6 2.f3 Qh4# 1.g4 e5 2.f4 Qh4# 1.g4 e6 2.f4 Qh4#


5

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 ...


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