Keep The Difference Small

Filling a \( 5 \times 5 \) grid with distinct integers from 1 to 25, what is the minimum of the maximum of the differences between 2 squares that share a common side?

\(\)
Hint: As an explicit example, for the naive arrangement below

the maximum of the differences between 2 squares sharing a common side is equal to 5. This is because, in the above table which is one of all \(25!\) possible arrangements, the maximum never exceeds 5, as can be seen by the differences between many neighboring cells: \[6-1=7-2=\cdots=24-19=25-20,\] which are all equal to 5. Hence, the minimum of the maximum we are looking for is at most 5.

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