Small chess problem

Besides the well-known knight problem and queen problem, there are many other exciting questions in the world of chess. I have touched on two small curiosities in a previous blog entry. If you deal with chess problems mathematically, you will soon realize that mathematics provides very simple and enlightening answers to many questions.

As an example, I will now treat the following problem: You look at an empty, regular chess board with 64 fields and place a white queen at any position \((x,y)\) . How many possible moves does the queen now have?

Using the symmetry properties of the board, we transform each point \( (x,y) \in \{1,2,3,4,5,6,7,8\} \times \{1,2,3,4,5,6,7,8\} \) into its counterpart \( (x',y') \in \{1,2,3,4\} \times \{1,2,3,4\} \) in the lower left quadrant and choose the minimum \(z\) of the two coordinates. Finally, we obtain \(7\) horizontal, \(7\) vertical and \( 7 + 2\cdot(z-1)\) diagonal pulling possibilities, thus:

\[ f:\{1,2,3,4,5,6,7,8\} \times \{1,2,3,4,5,6,7,8\}, \\ f(x,y) = 2 \cdot \min(-|x-4,5|+4,5; -|y-4,5|+4,5)+19 \]

The inclined reader can easily extend the problem to chessboards of size \(n^2\).