# Polyiamond 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 polyiamonds.
A polyiamond with *k* cells requires at least *k* colors.
In the diagrams below, color counts that exactly meet this requirement
appear in red.

For polyhexes, see Polyhex Variegation.

## Diamond

For the diamond two colors suffice:

## Triamond

For the triamond four colors are needed, to distinguish the cells
of a triangle with side 2:

## Tetriamonds

The same pattern admits two of the three tetriamonds:

The third tetriamond requires 6 colors, to distinguish the cells
of a hexagon with side 1:

## Pentiamonds

The same pattern optimally admits the I, Q, and U pentiamonds:

The fourth pentiamond requires 8 colors:

## Hexiamonds

The pattern for the V tetriamond also admits five hexiamonds:

The pattern for the J pentiamond admits five other hexiamonds:

Here is the best pattern I have found for the remaining two hexiamonds.

Last revised 2018-07-19.

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

Col. George Sicherman
[ HOME
| MAIL
]