Enigma 69: Maximum Queen moves
10 March 2013
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From New Scientist #1212, 31st July 1980 [link]
If you place Queens (as many as you like) on an ordinary 8 × 8 chessboard, in such a way that you have the greatest possible number of moves open to you, you will find you cannot do better than put a Queen on each border square. You thus use 28 Queens in all. Each of the 36 unoccupied squares can then be moved to by just 8 Queens. Total moves available: 36 × 8 = 288. And you cannot achieve 288 with fewer than 28 Queens.
Suppose I now ask you to do the same on a 4 × 4 board. If you place a Queen on each border square, you will as the diagram shows, achieve 4 × 8 = 32 possible moves. Can you do better? Remember that your prime purpose is to maximise the number of possible moves; subject to that you want to use as few Queens as you can.
How many possible moves can you get on a 4 × 4 board? And how will you place your Queens?