# Hexiamond Minimal Covers

December 16th, 2010 by munizao

I define a cover of a set of polyforms as a shape or shapes into which each polyform in the set can fit. I’ve written up an exploration of related problems on my recreational mathematics non-blog. Most of these problems use pentominoes or other polyominoes; it lately occurred to me that I had done a disservice to the hexiamonds, which are just as worthy of attention. So here’s a minimal (ten cell) cover of the twelve hexiamonds:

A proof of its minimality is simple. There are two ways that the bar and hexagon hexiamonds can be superimposed with maximal overlap:

Each of these contains nine cells, and neither is a cover of all of the hexiamonds. (The one on the left is missing the butterfly and chevron hexiamonds, the one on the right, just the butterfly. ) Therefore any cover must contain at least ten cells, so the one given above is minimal. (The nomenclature I’m using for individual hexiamonds is given on this MathWorld page.)

In fact, there are five ways to complete a cover by adding a single triangle (marked in blue below) to one of the figures above:

And these are the minimal covers produced:

The animation at the top of this post displays a cycle where a single cell is moved at each step. (I posted about this previously with the pentominoes in a minimal cover.) The hexiamonds have the handy property that their number is exactly twice their size, which leads naturally to the following problem:

Problem #16: Starting with a hexiamond with all cells labeled, find a sequence of single cell moves that cycles through all twelve hexiamonds, returning to the first hexiamond, and that moves each labeled cell exactly twice. Bonus points if the set of cells used is either a minimal cover or a 2×3 parallelogram. Bonus points as well for satisfying conditions like those of problems #10 and #11 in the post linked above. (All cells return to their starting positions, or all cells end up in a cyclic permutation of their starting positions.)