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The integers 1 to 4 are positioned in a 6 by 6 square grid as shown and cannot be moved. The integers 5 to 36 are now placed in the 32 empty squares. Prove that no matter how this is done, the integers in some pair of adjacent squares (i.e. squares sharing an edge) must differ by at least 16.

\begin{array}{|l|l|l|l|l|l|} \hline & & & & & \\ \hline & 1 & & & 2 & \\ \hline & & & & & \\ \hline & & & & & \\ \hline & 3 & & & 4 & \\ \hline & & & & & \\ \hline \end{array}

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1 Answer 1

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Perhaps I'm missing something but here is a quick proof

There are 16 distinct squares which are adjacent to either 1,2,3 or 4. This means that at least one of these squares will contain an integer which is greater than 19. Hence, one of 1,2,3 or 4 will be adjacent to an integer at least 16 greater.

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    $\begingroup$ Very nicely done. $\endgroup$
    – graffe
    May 20 at 10:47

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