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?
ANSWERS
Solutions were received from Joe DeVincentis, George Sicherman, and Maurizio Morandi.
Decreasing Square Paths
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

n=10
40 paths (JD)
 n=11
50 paths (GS)
 n=12
67 paths (GS)
 n=13
81 paths (MM)

n=14
97 paths (MM)
 n=15
108 paths (MM)
 n=16
116 paths (MM)

Decreasing Equilateral Triangle Paths
n=1
1 path
 n=3
2 paths (GS)
 n=4
6 paths (GS)
 n=5
6 paths (GS)

n=6
9 paths (GS)
 n=7
13 paths (GS)
 n=8
19 paths (GS)
 n=9
27 paths (MM)

If you can extend any of these results, please
email me.
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