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Problem of the Month (March 2017)

Consider tilings of an n×n square with integer squares, where there is a unique largest square of side k. Define a decreasing square path to be a sequence of k squares, of sides k through 1, where each pair of consecutive squares touch along an edge. What tiling of an n×n square maximizes the number of decreasing square paths? In particular, what is the largest possible number of decreasing square paths of length k in tilings of arbitrarily large squares? What are the answers for triangle tilings? What about other reptiles?

n=1

1 path
n=3

4 paths
n=4

8 paths
n=5

8 paths
n=6

14 paths
n=7

24 paths
n=8

30 paths
n=9

36 paths


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