25 November 2011


It all started early on the morning of November 26th, 2010. To the day one year ago, my first blog post was published. My original intention was to post every third day. This worked quite well for a while, but I couldn't keep up the pace. Of course, the texts don't always express deep thoughts. Still I aim to maintain a certain quality level and avoid too much blabber. Selecting a subject, thinking about the wording and the layout, etc. — that often takes some time.

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.

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