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#1 2015-06-30 15:07:18

phanthanhtom
Member
Registered: 2012-06-22
Posts: 290

Place 32 knights on an 8x8 chessboard

It is proven that we can place a maximum of 8 queens, 8 rooks, 14 bishops, 16 kings or 32 knights on an 8x8 chessboard such that no two pieces can attack each other. (let's not talk about pawns here)

For 8 queens, according to Wikipedia, there are 12 fundamental solutions (rotations and reflections count as one solution only) and 92 distinct solutions (rotations and reflections create new solutions). What are the corresponding numbers for placing 32 knights? (You might also want to try rooks (apparently thousands), bishops and kings, but those are probably larger numbers).

Also, if we stick the two opposing edges of the chessboard together to make a cylinder (without bases), how many queens can we place at most on the board? (so that there still is no pair of attacking queens)

Another way to place them is:

Divide the 8x8 chessboard into 16 2x2 subboards. Colour these subboards with alternating colours (black/white etc). Then you can place 32 knights on these new dark/light squares and that's two new solutions.

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