by byakuugan » Sat May 07, 2011 9:24 pm
Does anyone know how I could obtain a program that can calculate tablebases where other pieces are treated as kings, like a 2-man tablebase where the goal is simply to stalemate/checkmate the other piece. A simple example would be Queen vs. Knight with no kings on the board, there is a simple algorithm the queen uses to force checkmate against the knight on any size board, which seems like it would be simple to teach a computer, and maybe even find a formula for determining the exact number of moves it will take.
It seems more logical to start out with 2-man endings, since many of these endings are simply repeating the same formations over and over again, like how Wazir vs. Wazir is a win for whoever has the opposition, and how Ferz vs. Same-Colored Ferz is a win for whoever has the opposition. In fact, any "leaper" chess piece with 4 moving paths can stalemate another piece of the same type. Leapers with 8 moving paths can escape each other. I am not sure if there is another terminology to describe the number of symmetrical moving paths a piece has, but I use "octosymmetrical" and "quadrisymmetrical"
I figured out why all pythagorean triples can be defined by a chess piece, but it would probably take a 10-part video series to describe proof of the theorems. I will make this video series as soon as I get the proper equipment to actually film myself drawing on a board.
Short description:
All "octosymmetrical" leapers, like knights, camels (1,3), giraffes (1,4), zebras(2,3), etc. have 3 corresponding quadrisymmetrical leapers, by taking away half of the piece's moves. The aura of the piece will now be a rectangle rather than an octagon.
The 3,4,5 pythagorean triple is the KNIGHT TRIPLE, since the 3 quadrisymmetrical knights are colorbound, 3, 4, and 5-ways. I call these pieces DiagoKnight, StraightKnight, and SquareKnight.
If x and y represent the movement of a leaper, then the 3 formulas are:
2xy determines the colorboundness of a quadrisymmetrical leaper whose aura is a rectangle with sides that are ranks/files
(x+y)(|x-y|) determines the colorboundness of a quadrisymmetrical leaper whose aura is a rectangle with sides that are diagonals
xx+yy determines the colorboundness of a quadrisymmetrical leaper whose aura is a square.
It is easy to figure out that (2xy)^2 plus (x+y)(|x-y|)(x+y)(|x-y|) equals (xx+yy)^2, but it would take a lot of animation to provide a visual representation of why it works. I am not sure how pythagorean triples would help in the evolution of understanding tablebase endings, but at least it is a start.