Problem 01

Problem 01

In a 4 x 6 crate, we put k bottles, such that every row and column contains an even number of them. For which values of k is this possible?

Solution

Clearly, k must be even, as the sum of even numbers is even. If k is a multiple of 4, several 2 x 2 squares easily yield a good example. For k = 2 and k = 22, it is clearly impossible. Therefore, one would expect that it works if and only if k ≡ 0 (mod 4).

This is however false. The following example demonstrates that it is possible for k = 6, and by adding one or two 2 x 2 squares or by leaving these 6 spots open, one finds examples for k = 10, k = 14 and k = 18 as well.

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Remarks

While not related to any theory, it is a nice problem for their general problem solving abilities, as well as a fun puzzle for their non-mathematical friends, physically puzzling with 6 bottles and a crate. Popularizing mathematics starts in your own class!