Problem of the Month (October 2015)

The December 2001 Math Magic problem concerned tiling squares with strips of squares of sides 1, 2, ... n. This month we ask a similar question. Which triangles and regular hexagons can be tiled with strips of equilateral triangles of sides 1, 2, ... n?

ANSWERS

Submissions were received from Berend Jan van der Zwaag, George Sicherman, Joe DeVincentis, Mark Thompson, Bryce Herdt, Maurizio Morandi, and Jeremy Galvagni.

The known triangles (with side s, formed by strips of triangles of sides 1-n) are shown below:

n\s123456789 101112131415161718192021222324252627282930313233343536
1 EF
2 EF
3 EF EF
4 EF EF EF EF
5 EF EF EF EF BZ EF EF
6 EF EF EF EF EF EF EF EF EF EF EF
7 EF BZ EF EF EF EF EF EF EF EF EF EF EF EF EF EF
8 BZ GS BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ BZ
9 GS GS GS GS GS GS GS GS GS GS GS GS GS GS JD JD JD JD JD JD
10 ? ? ? GS GS GS JD GS GS GS GS GS GS GS GS GS GS
11 ? GS ? GS GS GS GS GS GS GS GS GS GS GS
12 ? ? ? ? GS GS GS GS GS GS GS

n\s373839404142434445464748495051525354555657585960616263646566
9 JD JD JD JD JD JD BH JD JD
10 GS GS GS GS GS GS GS MM GS MM MM GS GS MM MM ? ? ? MM
11 GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS ? GS ? ? ? ? ? ? GS
12 GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS

Jeremy Galvagni noted that a solution is always possible for s=n(n+1)/2 by mimicking the solutions for n=8 and n=9.

The known hexagons (with side s, formed by strips of triangles of sides 1-n) are shown below:

n\s3456789 101112131415161718192021222324252627282930313233343536373839
3 EF
4 EF
5 EF GS GS
6 EF GS GS GS GS
7 EF GS EF GS GS EF ?
8 GS GS GS GS EF EF MM EF EF ? EF
9 GS GS GS EF ? EF EF EF ? EF ? EF EF ?
10 GS GS EF ? ? EF EF ? EF EF EF EF EF ? EF ? ? EF
11 GS ? ? ? EF EF EF EF ? EF EF EF ? EF EF EF EF EF ? EF ? ? EF
12 ? ? ? ? ? EF EF EF EF EF EF EF EF EF EF EF EF EF EF ? EF EF ? EF EF ? ?

Mark Thompson sent this larger example with n=15 and s=60.

George Sicherman wondered about polyabolo strips. Bryce Herdt noted that any equilateral triangle solution can become a polyabolo solution by affine transformation.

n\s123456789 101112131415161718192021222324252627282930313233343536
1 GS
2 GS
3 GS GS
4 GS GS GS GS
5 GS GS GS GS GS GS GS
6 GS GS GS GS GS GS GS GS GS GS GS
7 GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS
8 GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS ? GS

This made me wonder about polyabolo strip tilings of squares and octagons.

n\s123456789 101112131415161718192021222324252627282930313233343536
1 GS
2 GS
3 GS GS GS GS
4 GS GS GS GS
5 GS GS GS GS GS GS
6 GS GS GS GS GS GS GS GS GS GS ? ?
7 GS GS GS GS GS GS GS GS GS GS GS ? GS ? ? ? ? ?
8 GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS GS ? ? ? ? ? ?

Bryce Herdt sent the first solution for octagons with n=9 and s=12.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/20/15.