# Small chess problem1114

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 $$n^2$$.

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