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I have been working on the following problem without much success:

Counters are placed on 25 of the squares of a 6 × 7 chessboard. Prove that there exists a 2 × 2 subboard with at least three counters on its four squares.

This problem comes from the book "Grade Five Competition from the Leningrad Mathematical Olympiad 1979–1992".

All I have been able to deduce is that if we color the grid pink and white in the following way:

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then I can conclude that (by the Pigeon-hole principle) that at least one counter must be on a white square. But I can’t solve any further.

I am not looking for hints or answers because the book has that information. I am not ready to give up on solving this problem.

What I am looking for is related and hopefully simpler problems (with answers hidden by spoilers) like this so I can solve them and eventually return to my book problem.

Will Octagon Gibson
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A natural source of the problems you are looking for (and which can provide some general intuition also for the much wider class of problems) is to consider small instances. Namely, given a small $m\times n$ board, which is the smallest number of its squares which can be settled by counters, such that there always exists a $2\times 2$ subboard with at least three counters on its four squares.

Alex Ravsky
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