The fewest queens that guard the whole board

Question

What is the smallest number of queens that can be placed on an 8 by 8 chessboard so that every square is either occupied by a queen or attacked by one? Give an arrangement, and explain why fewer cannot work.

Accepted Answer

Five queens. Why four are not enough. A queen in the best position (near the centre) attacks 27 squares and occupies 1, covering at most 28. Four queens would cover at most 112 squares if nothing overlapped, which exceeds 64, so the simple count does not settle it. What settles it is that the lines of attack of four queens, at most 4 rows, 4 columns and 8 diagonals, always leave some square uncovered; an exhaustive check of all 635,376 ways to place four queens confirms that no four queens cover the board. The check was first done by hand in the 19th century and takes a computer well under a …

The fewest queens that guard the whole board

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