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Problem of the Month
(October 2019)

We say the signature of a tiling of integer squares in a rectangle is a total ordering of the set of integers that count the number of horizontally and vertically adjacent squares for each square in the tiling. For example, 1=3>2 is the signature of a tiling that has the same number of squares with 1 neighbor and 3 neighbors, which is more than the number of squares with 2 neighbors.

Given such a total order, what is the smallest (in terms of area, then number of squares) tiling with that signature?

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