Problem #11

In an n×n array of unit squares, two squares are said to be adjacent if they share an edge.

• What are the maximum and minimum values that can be attained by placing the integers 1 through 9 inclusive into a 3×3 array and taking the sum of the differences of the values in adjacent squares?

  For example, for a 2×2 array, there are essentially only three cases to consider as shown below. In this case, the minimum value is 6 and the maximum value is 8.

  

• What's the solution to the analogous problem for a 4×4 array?

• What's the solution for a 2×2×2 array of cubes where two cubes are adjacent if they share a face?

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