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December 17

Knights problem

What is the optimum placement of knights on an indefinitely large chessboard (ignoring edge effects) so that every square (including squares on which the knights are placed) is attacked? "Optimum" means number of knights divided by total number of squares is minimised. When I first conceived of this question I assumed that the answer would be a repeating pattern but now I'm not even sure about that. — Preceding unsigned comment added by 86.151.119.39 (talk) 14:38, 17 December 2013 (UTC)[reply]

You can find a placement where every square is attacked precisely once, giving you a knight/square ratio of 1/8, the best possible. Here's one example: place a knight on a square, then move down 1 and place a knight on that square. Then move down 1 and right 2 and repeat. This will cover a diagonal band 8 squares high. You can then cover the entire board with repeated bands.--80.109.80.78 (talk) 15:11, 17 December 2013 (UTC)[reply]

Figure 6 in [1] gives another example. PrimeHunter (talk) 16:57, 17 December 2013 (UTC)[reply]

Ah, thanks, I thought the answer was more complicated. — Preceding unsigned comment added by 86.151.119.39 (talk) 20:32, 17 December 2013 (UTC)[reply]