# Polyiamond Exclusion

## Introduction

In the 1950s, Solomon W. Golomb investigated the question:
how few cells can you remove from the plane
to exclude the shape of a given polyomino?
Here I investigate the related question:
how few cells can you remove from the plane
to exclude the shape of a given polyiamond?

## Specific Results

Here are some patterns for small polyiamonds.
To exclude the diamond you must remove at least 1/2 the cells:

Here is the best known exclusion of the triamond.
If you find a better one please let me know.

If triamonds are excluded, every solid cell must be adjacent to at least
two holes, giving it 2/3 share in holes.
So a lower bound for the hole ratio is (2/3 /[1 + 2/3]), or 2/5.
There is a pattern with hole ratio 2/5 that excludes the triamond,
but it is not planar!
It covers the surface of a regular icosahedron:

This pattern optimally excludes two tetriamonds and two pentiamonds:

This pattern optimally excludes the V-tetriamond and the U-pentiamond:

This pattern is the best I have for the I-pentiamond:

This pattern of holes excludes five hexiamonds:

Here are my best results for the other hexiamonds:

For good measure here is a heptiamond exclusion,
not necessarily optimal:

## General Results

If a polyiamond of order *n* tiles the plane,
you must remove at least 1/*n* of the cells, one for each tile.
Any straight
polyiamond of even order *n* can be excluded with 1/*n* holes.
You can generalize
the following patterns to any *n* that is twice
an odd number by adjusting the distance between the columns of holes:

Similarly, you can generalize
the following patterns to any *n* that is twice an even number.
Each column with holes contains *n*/4 upward holes, then
*n*/4 downward holes, and so on.

Straight polyiamonds of odd order require
more than 1/*n* holes.
If 1/*n* holes sufficed, then every row would need exactly
1/*n* holes, for if some rows had more than
1/*n*, others would have less than 1/*n* to compensate,
which would admit a straight polyiamond of order *n*.
Therefore every row must have a hole just at every *n*th cell.
But if *n* is odd, this is impossible, as the following diagram
illustrates:

Any regular hexagon can be excluded efficiently:

The same is not true of equilateral triangles:

## Optimality Proofs

The next diagram demonstrates the optimality of some of the exclusions
with more than 1/*n* holes.
You need at least 2 holes in the tiles in the bottom row
to exclude the figures in the top row:

Back to Polyform Exclusion, Equalization,
Variegation, and Integration
<
Polyform Curiosities

Col. George Sicherman
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