Recommendations for chess themed math exercises

Question

I am trying to organize a recreational math class for a group of high school students (mixed years), themed around the game of chess. Ideally, I would like to prepare exercises that simply require a basic knowledge of chess rules.

What I have found so far tend to be exercises more on the advanced side of mathematics (in particular the book "Mathematics and Chess" by Petkovic), but e.g., in the book of "Finite Markov Chains and Algorithmic Applications" by Olle Häggström:

Random chess moves. (a) Consider a chessboard with a lone white king making random moves, meaning that at each move, he picks one of the possible squares to move to uniformly at random. Is the corresponding Markov chain irreducible and/or apreiodic?

This is a really cool and simple exercise, both from the chess aspect (as now we can repose the question for other pieces and compare them), and from the perspective of mathematics it constitutes a simple and accessible example for teaching about reducibility and periodicity of markov chains. Although the latter would be too advanced a topic for my students, the problem nonetheless showcases the kinds of exercises I am looking for, where solving it boils down to solving a little chess puzzle.

• Any resource recommendations (lecture notes, books, websites, anything) on mathematical concepts being taught through chess themed exercises would be perfectly admissible, specially if the level is generally accessible for high school students.
• Alternatively, if you simply have an idea for an exercise, please feel free to also post as an answer, it would be much appreciated!

Answer 1

Retrograde analysis is an engaging use of recreational logic applied to chessboards. The typical gateway is two books by Raymond Smullyan, "Chess Mysteries of the Arabian Knights" and "Chess Mysteries of Sherlock Holmes". (They are companion books, where the easy and challenging problems were divided between them.)

In a high school math elective context, this would be a strong example of proofs and deductive reasoning. Instead of the usual framing of the axioms of Euclidean geometry, the "axioms" are the rules of chess. Instead of proving triangle congruence, the goal is to prove what the last $n$ moves of the game were, or that White can't castle from a certain position, or that a piece on the board is promoted. As the example below shows, there is a lot of room for conjectures that can be proved or refuted, and a lot of multiple steps and false starts that can encourage group thinking and collaboration. And the presentation of an answer, either written or verbal or multimedia, can develop a student's strength in effective communication of complex ideas. These are the sorts of soft skills that can be invaluable for collegiate and professional work, especially in STEM fields.

Another thing that may be easier to do with retro analysis instead of Euclidean geometry is the ability for students to create their own problems after seeing a few problems of a certain theme and challenge their classmates to solve it or find a counterargument. Those acts of creation and judging are at the top of the higher-order thinking chain, which can be difficult to authentically develop in high school mathematics.

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Here is a sample problem (problem from Smullyan's "Chess Mysteries of Sherlock Holmes", solution by me) on the higher end of the difficulty scale for those two books.

The White King was accidentally knocked off the board. Given that every move in the game was legal, which square does it belong on?

It is either White’s move or Black’s move. Let us assume for the moment that it is White’s move. Then the Black king cannot be checked by the White bishop, so the White king must be blocking the White bishop. Two kings cannot be in adjacent squares, so the White king must be at b3. Therefore, the White king is under double check from the Black rook and the Black bishop. But there is no way that one of the Black pieces currently on the board could have moved to discover that double check. So our assumption that it was White’s move must be wrong.

Now that we know that it is Black’s move, we know that the White king cannot be in check. Therefore, the White king cannot be at b3 or c2. So the White bishop is checking the Black king. What was White’s last move that administered this mate? It couldn’t be the White bishop moving from b3 or c2, because then Black would have been in check at the beginning of White’s move. Could it be that the board is rotated 90 degrees and White’s last move was moving a pawn to the eighth rank and promoting it to that White bishop? Sneaky, but no; if the board were rotated 90 degrees, the lower right square would be black, which is illegal. The only remaining possibility is that the check was discovered by the White king moving from b3. That puts us under the same quandary of how the White king could be moving away from a double check from the Black rook and Black bishop, except that now we have the potential that the White king is moving to capture a Black piece that moved to discover the check. Is it possible for Black to move a piece on the last move to simultaneously discover a check from both the rook and bishop? Yes, under one circumstance: if a Black pawn moved from b4 to c3 to capture a White pawn en passant that was sitting on c4! Therefore, the White king’s last move was capturing the Black pawn on c3, so that is the only legal square for the White king to rest on.

To see a sequence of moves to get to this position, add a Black pawn at b4, a White pawn at c2, and move the Black bishop to a8 (although any square on that main diagonal would work as well). The last four moves in the game would be as follows.
… B-d5 check
P-c4 PxP en passant check
KxP check

Answer 2

This is a very interesting idea as I'm both a chess and math fan. Sadly, I don't know of any resources; maybe John Nunn has written something about it (since he was an excellent mathematician and chess player)?

Besides the knight's tour and n-queens, which I'm sure you're familiar with, a few puzzle ideas are coming to mind.

• How many moves can a given piece make before visiting a square it's already visited?
• How does the maximum length-game relate to the board size (i.e. if we played on a 10x10 board, would games last longer)?

I'm going to investigate and try to come up with more ideas. I hope this helps.