2005 was the year that the DaVinci code craze hit Belgium. (I started reading Dan Brown’s Digital Fortress and Angels and Demons a year before on the way back from a Warwick conference and when I read DVC a few months later it was an anti-climax…). Anyway, what better way to end 2005 than with a fitting chess problem, composed by Noam Elkies

The problem is to give an infinite sequence of numbers, the n-th term of the sequence being the number of ways White can force checkmate in exactly n moves. With the DVC-hint given, clearly only one series can be the solution… To prove it, note that White’s only non-checkmating moves are with the Bishop traveling along the path (g1,h2,g3,h4) and use symmetry to prove that the number of paths of length exactly k starting from h2 is the same as those starting from g3…

If that one was too easy for you, consider the same problem for the position

Here the solution are the 2-powers of those of the first problem. The proof essentially is that White has now two ways to deliver checkmate : Na6 and Nd7… For the solutions and more interesting chess-problems consult Noam Elkies’ excellent paper New directions in enumerative chess problems. Remains the problem which sequences can arise on an $N \times N$ board with an infinite supply of chess pieces!

Elkies, symmetry