Christian Hesse explained the theory of such logical puzzles in his book "Achtung Denkfalle" (English: "Careful: thought trap"):
The Zen Chess Puzzle illustrates, in chess context, the idea that knowledge is not merely knowledge. Knowledge about a certain fact in a group of two people can mean that each person knows the fact. But it can also be the case that each person also knows that the other person knows the fact, and even that each person knows that the other person knows that the other person knows the fact. This is possible not only for knowledge, but also for the lack of knowledge and even for the impossibility of knowledge of a certain fact.
Our Zen Chess Logical illustrates that a transition can take place from the impossibility of knowing something (namely which piece the Zen Master is talking about) to the possibility of knowing this, paradoxically by someone merely stating the impossibility of this particular knowledge.
If you have difficulty with the solution to our Zen Logical, here's a little warm-up task that might help you understand the process:
Two logicians go to a cafeteria. The waitress comes to their table and says: "Coffee for both of you?" The first logician says "I don't know," and the second says "Yes." What does each want, and how did they figure it out?
The Zen Chess Logical
The logical problem we published two weeks ago was created by Christian Hesse, and is typical for his way of thinking. A Zen master visits a chess club and sees two students studying a chess position with the following material on the board:
The Zen master thinks of one piece and whispers some information into each student's ear. He tells Kaito which piece type he is thinking of, and Toshi the colour of the piece. Toshi is the first to react: "There is no way either of us could deduce which piece you mean." To which Kaito says: "Okay, now I know which piece it is!" Toshi: "Aha, then I know as well."
The chess logical was a task for our readers: find out which piece the Zen master meant, and how Kaito and Toshi could deduce it. So here's the solution:
- Both Kaito and Toshi know that Kaito has been told the type and Toshi the colour of the piece.
- Toshi immediately says nobody can know which one it is.
- From this Kaito deduces it must be white, because otherwise Toshi had to consider the possibility that it might be a knight or a queen, in which case Kaito would know which piece the Zen master was thinking of.
- Since Kaito now says he knows which piece it is, Toshi deduces it must be the white king, because bishop or pawn would not specify the exact piece. So now Toshi also knows it was the white king.
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A number of readers got it right, while we can tell you that so far many world-class players have been unable to find the answer to the Logical. With that they have helped us collect data for our attempt to answer the question: Are chess super-talents generally smarter than regular kids? So far only two young talents have given us the correct answer. And grown-ups? John Nunn solved it almost immediately, Ken Thompson too, followed by Vishy Anand. But not many more.
Professor Christian Hesse with one of the successful solvers
The Yakimchik study
The second task for our readers was to solve the study on the chessboard. It was given to us by Dr Karsten Müller (to match the Chess Logical material) and is cleverer than it looks at first sight. Here is the full solution, nicely annotated by our problem expert Siegfried Hornecker:
Have you seen this kind of drawing mechanism before? We hadn't.
