# Polyhex Variegation

## Introduction

In his paper Polyominoes on a Multicolored Infinite Grid

(in Thane Plambeck and Tomas Rokicki, eds.,
*Barrycades and Septoku: Papers in Honor of Martin
Gardner and Tom Rogers,*
Providence, 2020, MAA Press, Spectrum Series, v. 100, pp. 29–36),
Hans Hung-Hsun Yu investigates how many colors are needed for
the cells of the plane to ensure that a given polyomino has no
two cells of the same color.
Here I consider the corresponding problem for polyhexes.
A polyhex with *k* cells requires at least *k* colors.
In the diagrams below, color counts that exactly meet this requirement
appear in red.

For polyiamonds, see Polyiamond Variegation.

## Dihex

The dihex requires three colors:

## Trihexes

For the A and I trihexes the same pattern suffices:

The third trihex requires four colors:

## Tetrahexes

The same pattern admits three of the seven tetrahexes:

The I tetrahex requires five colors:

The Q and U tetrahexes require seven colors:

The J tetrahex requires nine colors:

## General Result

A straight polyhex with an odd number of cells
requires only as many colors as it has cells.
The diagram below illustrates the typical pattern.

*Last revised 2018-07-19.*

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

Col. George Sicherman
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