4x4 chess

[byakuugan]

I was studying a lot of 4x4 endgames on fairygen by multiplying all the chess piece movements by 2, (Kings become Alibabas, Knights become 2,4-leapers, Bishops become alfilriders etc.) and was wondering if anyone knew how to generate tablebases for larger chessboards. It would make sense to try to generate tablebases for the largest chessboard possible, since smaller boards can be studied simply by multiplying the piece movements by a certain number (you could also generate 3x3 endgames by multiplying piece movements by 3), but since not all board sizes can be generated using this method, there must be a "friendly" set of board sizes with the most potential to study smaller boards on it.

I made a playlist of a specific 2-leaper mating pattern on 8x8 boards
http://www.youtube.com/watch?v=hx0UtD6o ... re=mh_lolz

It would be interesting to learn the board sizes where each set of leapers lose their mating potential, or if there are algorithms to know what range of boards mate is possible on, since checkmate for a certain endgame may be possible on a 1000000 by X sized chessboard (where X is small number), but not on an X+1 by X+1 sized chessboard, where the extra line of freedom is all the defender's king needs to hold a draw.

[h.g.muller]

I guess fairygen could be easily adapted to do true 4x4 generation, saving a factor 4 on memory usage (and run time) per chess man. Even with 7 men it would then only use 1/4 of the memory it uses now for a 5-men, and 8-men (octo-symmetric) would only take 512MB. But of course 4x4 boards are not very interesting.

I once did make a version of fairygen that used a 16x16 board. Fairygen is based on the assumption that the board dimensions are powers of 2, so it was the next-higher size, but I made it such that the 16x16 board could be partly filled up with immovable uncapturable pieces, to simulate smaller boards. To preserve octo-symmetry, however, it only could do even x even. The big board made it quite memory demanding: for 3-men it required slightly over 2MB, which was still quite doable, but for 4-men it would already over 512MB, which was more than the computer I had at the time could handle. (Especially since I was mainly interested in 10x8 boards, doubling the memory requirement by breaking diagonal symmetry.) So I used it only to calculate 3-men end-games on various boards, see http://www.chessvariants.org/index/list ... _checkmate .

During my holiday I made a symmetry-less generator for 3+1-men on arbitrary board size, which is more memory friendly in the sense that it does not use needlessly large board. (But the fact it doesn't exploit symmetry could cost you a factor 8, of course.). As it was only for determining mating potential of pairs of pieces that would have no mate positions on their own, it does not care about sub-end-games, and assumes that any legal capture by black immediately draws, making life very simple. I posted it on TalkChess: http://www.talkchess.com/forum/viewtopi ... 1&start=10 .

The board size can be easily set, but also there the memory demands quickly grow with board size, and on the small netbook I had with me it could not allocate enough memory to do 12x12. I could do KBNK on 10x10, however. I did not try it on my larger machine yet. It is a very 'bare bones' generator; it just prints the stats, and there is no probing code included, so there is no way to play against it. But it is good enough to see if there is mating potential, and there are no requirements on the symmetry at all. (But it does not implement promotion, so it would still not work with Pawns...)

[byakuugan]

It seems that all chess-related concepts are encrypted concepts of other aspects of the game. Every endgame solved also solves an infinite number of related problems. Instead of 4x4 endings, I used a similar method to generate endgames on a 32-square board with diagonals as edges instead of ranks/files.
Kings become Ferz/Dabbaba hybrids, Rooks become Bishops, Knights become Camels, etc. then when translated to the 32-square chessboard they revert back to original chess pieces. The mating potential of leapers is a lot different on this chessboard.

[h.g.muller]

For some reason I cannot even run the 12x12 board with 4 men on my bigger machine, although it would only take 410MB. With 11x11 all corners are the same color, so of course nearly half the positions have the Bishop on the wrong color in KBNK, so that the number of lost positions drops from over 80% at 10x10 to around 40%. Of course the positions that are still won can now be won much more easily, because any corner will do. And in 3 vs. 1 end-games it is usually driving the king from the safe corner to the deadly on that fails first. So the largest DTM on 11x11 is only 38, while at 10x10 it was 47. Quite possible that 14x14 would always a draw, but 15x15 and 17x17 would be a win, provided you had the right Bishop.

[byakuugan]

Perhaps an efficient way to figure the range of board sizes where a specific checkmate would be possible, would be to figure an algorithm based on rules where each player must move ALL their pieces on their turn. The reason is that with these rules, most mates with king + 2 pieces vs. king can be forced on any size chessboard, so there would be algorithms to determine the best moves in any given situation. Studying these algorithms on large chessboards, it would then be possible to understand why the endgame would not be winning with regular chess rules (the defending king can exploit the pieces' lack of speed when they must move one at a time) and it might then be possible to develop simple formulas for determining the maximum corralling potential of pieces with regular chess rules.