I've never advertised this blog a lot as I write just for the fun of it. Hence, the themes I give attention to might not always attract a big audience. But not earning money with the page hits anyway, I don't really care. However, in the course of time, the number of visitors increased and that's surely an encouragement to keep on posting. One interesting fact is that a lot of people are interested in detective chess (top search keywords).
Today, I have a nice composition for you that combines chess and mathematics. It's taken from the book Schach und Zahl by Eero Bonsdorff, Karl Fabel and Olavi Riihimaa.
|Dr. Erkki Pesonen|
|Schach und Zahl, 1966|
|What is the most probable end of the game?|
All choices have the same probability.
a) consider all legal moves
b) first choose the piece, then the move
The key to the solution is this: The fewer moves the higher the probability. A closer look at the position tells that there are two ways to end the game in only three half-moves. Counting moves is all what's left.
a) 1. Nb4 b1=R 2. Nc2#.
Possibilities on half-move 1: Ka4, Kb3, Nb4, Nb8, Nxc5, Nc7 (6)
Possibilities on half-move 2: Kb1, b1=Q, b1=R, b1=B, b1=N, cxb4 (6)
Possibilities on half-move 3: Ka4, Na2, Na6, Nc2, Nc6, Nd3, Nd5 (7)
Thus, the probability is 1/6 * 1/6 * 1/7 = 1/252.
Thematic try: 1. Nxc5? b1=B 2. Nb3# with 1/6 * 1/5 * 1/11 = 1/330.
b) 1. Nxc5 b1=B 2. Nb3#.
Choices on half-move 1: 1. K, N (2) 2. Nb4, Nb8, Nxc5, Nc7 (4)
Choices on half-move 2: 1. K, P (2) 2. b1=Q, b1=R, b1=B, b1=N (4)
Choices on half-move 3: 1. K, N (2) 2. Na4, Na6, Nb3, Nb7, Nd3, Nd7, Ne4, Ne6 (8)
Therefore, the probability is 1/2*1/4 * 1/2*1/4 * 1/2*1/8 = 1/1024.
Thematic try: 1. Nb4? b1=R 2. Nc2# with 1/2*1/4 * 1/3*1/4 * 1/2*1/6 = 1/1152.