# Polycairo 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 polycairo?
If you find a more efficient exclusion, please write.

## Specific Results

Here are some patterns for small polycairos.
To exclude either dicairo you must remove at least 1/2 the cells.
This also holds for the two tricairos:

These tricairos are excluded optimally.
The tetracairos are probably optimal.

This pattern optimally excludes six tetracairos:

This pattern optimally excludes three tetracairos:

These tetracairos are optimally excluded by these patterns:

These patterns are the best known for excluding these tetracairos:

## General Results

If a polycairo with *n* cells tiles the plane,
you must remove at least 1/*n* of the cells, one for each tile.
## Optimality Proofs

The next diagram demonstrates the optimality of two of the exclusions
with more than 1/*n* holes.
The numbers show how many holes you need in the green figure
to exclude the yellow figure:

Last revised 2018-05-28.

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

Col. George Sicherman
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