Zucca's Problem

Livio Zucca's Tetrominoes Challenge Page challenges you to find plane regions that can be tiled with each of a given set of tetrominoes and no others. Here I show solutions to the corresponding problem for some other polyforms. If you have a smaller solution for any of these sets, please let me know.

Polyiamonds

A polyiamond is a figure made of equilateral triangles adjoined edge to edge. A tetriamond has four triangles and a pentiamond has five.

Tetriamonds

Pentiamonds

Tetrahexes

A tetrahex is made of four regular hexagons adjoined edge to edge. The September 2004 edition of Erich Friedman's Math Magic looks at pairwise compatibility for various polyforms, including tetrahexes and extrominoes.

There are 7 tetrahexes, so a complete solution for Zucca's problem for tetrahexes has 120 cases! That's too many for me. Here are solutions for all the pairs but one. They may not be minimal. Thanks to Dr. Friedman for improving on some of my original results.

Pairs

And here, by courtesy of Dr. Friedman or me or both, are some solutions for sets of three or more:

Triplets

Bigger Sets

Extrominoes

An extromino, or extended tromino, is made of three orthogonal squares joined edge to edge or kitty-corner. (Golomb calls it a pseudo-tromino.) The results below are Dr. Friedman's except where noted otherwise.

Tripents

Polypents don't look as neat as the previous polyforms, because they cannot form solid masses. Here is a compatibility diagram for the 2 tripents.

Dr. Friedman has worked out compatibilities for all 7 tetrapents. You can see them at his MathMagic Page on polyform compatibility.

Back to Polyform Curiosities.

Col. G. L. Sicherman [ HOME | MAIL ]