Knight combinatorics

Probability Level 3

In how many ways can we place 32 32 knights on an 8 × 8 8\times 8 chess board (at most 1 knight to each field), such that no two of them can attack each other?


Note: A knight's valid moves are shown by the stars in the picture.


The answer is 2.

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Kenny O.
Sep 2, 2017

When a knight moves, the Colour of the tile it is on changes, meaning that all the 32 knights must be on the same Colour tile. This is possible as there are exactly 32 black and 32 white tiles. As there are only 2 Colours and no tile is empty (other than the tile of the other Colour), there are only 2 ways.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

There is a flaw in your argument. You state that all the knights must be placed on the same color of tile with no explanation as to why that must be the case. Sure, intuitively it seems obvious, but you provide no proof for that. You have demonstrated that the 2 positions where all knights are on the same color work, but there needs to be some argument that you could not have, for example, 16 knights on white squares and 16 knights on black squares.

Kyle Coughlin - 3 years, 9 months ago

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