On each unit square of a 9×9 square, there is a bettle. Simultaneously, at the whistle, each bettle moves from its unit square to another one which has only a common vertex with the original one (thus in diagonal). Some bettles can go to the same unit square. Determine the minimum number of empty unit squares after the moves. Pierre.
Problem
Source: Me
Tags: analytic geometry
21.12.2004 19:37
I promise I won't spoil any more of these, but this problem is too cute : Give the squares coordinates: from (1,1) to (9,9). The squares with both coordinates odd can only be occupied by beetles from squares with both coordinates even. There are 21 squares with both coordinates odd and 12=21−9 squares with both coordinates even, meaning that at least 9 squares remain empty. It's very easy to find a set of movements leaving exactly 9 squares empty, so the answer is 9.
26.12.2004 14:58
Here is a solution from a 12-year old student...
This was a very nice little problem, Pierre!
26.12.2004 15:09
Thanks you. I agree. But, unfortunately, I'm not the creator of it. I just proposed it here from our pre-TST. Pierre.
26.12.2004 16:12
It is an old russian problem.
22.01.2005 14:13
colouring in 4 colours makes the solution show up easily (its equivalent to all other solutions here actually)