# Chess and mathematics

I have to choose a research-like project to follow the next year. Because I'm a chess enthusiast, I was thinking of trying to tackle an (open) problem related to chess, and relevant to mathematics.

Can you suggest some material on the "mathematical side of chess" and related problems?

• One thing that has puzzled me for quite some time: Given two perfect players $A$ and $B$, where $A$ has only a king, a knight and a bishop left, and $B$ has only a king left - can $A$ beat $B$? (although I suppose it's not sufficient for a research project). – barak manos Aug 17 '14 at 14:51
• @barakmanos: Yes, with a couple of exceptions. – Daniel R Aug 17 '14 at 14:53
• @DanielR: What I meant is - If $B$ is perfect, will he/she be able to avoid being check-mated (even if $A$ is also perfect)? – barak manos Aug 17 '14 at 14:57
• @barakmanos No, it takes at most $35$ moves to force a mate against a perfect opponent, unless the position is such that the weaker side can force a stalemate or capture a piece. All according to the link I posted above. – Daniel R Aug 17 '14 at 15:01
• @DanielR: Got it, thanks :) – barak manos Aug 17 '14 at 15:03

The Knights tour is a famous chess/math problem.

Start here.

• This answer was flagged as low quality. As link only answers are often undesirable, I marked the flag as helpful. That was converted to a downvote, which mechanism had somehow escaped my attention until now. I think Noam Elkies' page is one of the best sources (IIRC at least at one point he was working on a book about math problems in chess), so that was not my intention. Compensating with an upvote. Sorry about the mixed signals to all. – Jyrki Lahtonen Jan 2 '15 at 11:44

Writing a chess engine that plays only moderately well, is a considerable programming challenge. Imo, very few people have the discipline, ability and patience to actually organize and execute something like that from scratch, programming-wise.

There are some open sources which you can see and read, like that of GNU chess, Borland's Turbo chess and several others which have ended up public, yet it still is fairly difficult to even manage to improve some already existing engine.

I'd gladly consider it a fairly hot subject for a Ph.D. thesis.

• How much elo is only moderately well? Also, you agree that is even harder to create engine that plays better than a engine that plays worst? Lasy question to understand your post. – Santropedro Jan 16 '17 at 17:41
• Moderately well is in the range 1000-1600. Yes, I agree to your second question. The hard part with a chess engine is fine tune it to play in the 1900-2700 range. – Yiannis Galidakis Jan 17 '17 at 8:13
• Yes, I think we agree. I was asking you this: In my view (I haven't programmed chess engines ever, but I imagine works like this) If the elo of the engine is higher, it's harder to program. Someone could have a thepry that for some reason is hard to do "dumb" engines. However, seems to me you can just make it play some times the 5th best move, or the 10th best move (there are about 35 moves per position available, and those moves after 10 are rather bad) with random chance, and play good moves with some probability, so that you can make as low elo as you want.. – Santropedro Jan 17 '17 at 14:29

I studied rook polynomials. This problem asks how many rooks can be placed on a board in non-taking positions and has applications in scheduling.