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.


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Col. George Sicherman [ HOME | MAIL ]