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Problem 1216

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Combinatorics Difficulty 5.3 Prove it Ukrainian Mathematical Olympiad · Ukraine

Is it possible to fill in a table $7 \times 7$ with $24$ ones and $25$ zeros so that:
1) for each cell containing one, sum of numbers in neighbour cells equals one,
2) for each cell containing zero, sum of numbers in neighbour cells does not equal one?
(Two cells are neighbours if they have a common side.)

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Answer: Yes, it is possible (see the figure).

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.