# Polyhex 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 into the related question:
how few cells can you remove from the plane
to exclude the shape of a given polyhex?

## Dihex

The dihex is hard to exclude!
You must remove at least 2/3 of the cells:

## Trihexes

To exclude the bent trihex you can remove half the cells.
If you have a better exclusion, please let me know.

These trihexes and tetrahexes are excluded minimally.

## Tetrahexes

These tetrahex exclusions are minimal:

These are probably minimal:

## Pentahexes

This pattern is minimal for excluding the orange pentahexes,
and probably minimal for the others:

This pattern may be minimal for excluding the Y pentahex:

This pattern is probably minimal for exluding these pentahexes:

## Hexahexes

This exclusion for the straight hexahex is probably minimal:

## General Results

A straight polyhex of odd order *n* can be excluded with 1/*n*
holes.
Here is an example for *n*=5:

## Optimality Proofs

This diagram illustrates the optimality of some of the exclusions
with more than 1/*n* holes.
Every green figure tiles the plane.

*Last revised 2014-11-11.*

Back to Polyform Exclusion
< Polyform Curiosities

Col. George Sicherman
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