# Two-Pentomino Square Frames

• Introduction
• Nomenclature
• Table
• Minimal Solutions
• Balanced Variants
• Contiguous Variants
• ## Introduction

A pentomino is a figure made of five squares joined edge to edge. There are 12 such figures, not distinguishing reflections and rotations. They were first enumerated and studied by Solomon Golomb.

The January 2008 issue of Erich Friedman's Math Magic defined a frame as a square polyomino with a centered square hole. The problem was to find the frame with least area that could be tiled with a given polyomino.

Here I study the related problem of finding the smallest frame that can be tiled with two pentominoes. Thanks to Joyce Michel for suggesting this problem.

Bryce Herdt solved a balanced variant. Carl Schwenke and Johann Schwenke improved on some of my solutions. They also noticed that my balanced variant for pentominoes I and Z had fewer tiles than my general solution!

## Nomenclature

I use Solomon W. Golomb's original names for the pentominoes:

## Table

FILNPTUVWXYZ
F2881216128
I288128824828?864
L8888121281224816
N1288816128
P12888128121212812
T812812121216
U16241216812881212
V12881212886488
W281212126412
X?24121224
Y8888816128122416
Z641612816

## Solutions

So far as I know, these solutions have minimal area. They are not necessarily uniquely minimal.

## Balanced Variants

A balanced tiling has equal numbers of the two pentominoes. Here I do not show tilings from the previous section if they are already balanced.

## Contiguous Variants

In a contiguous variant, all the pentominoes with a given shape are connected at edges. Here I do not show tilings from the first section if they are already contiguous.

### 72 Tiles

Last revised 2024-02-27.

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Col. George Sicherman [ HOME | MAIL ]