# Cell Shifts for Polyominoes

## Introduction

Two figures can be tiled with copies of the same polyomino. The figures differ in only one cell. How near can the unmatched cells lie?

Over all such pairs of figures, a minimal vector from one unmatched cell to the other is called a minimal shift vector. Here I show minimal shift vectors for polyominoes up to order 5. Values in red are unproven. The nontrivial proven values are by Mike Reid, using Tile Homotopy Theory.

If you can solve any of the unsolved cases, please let me know.

For heptominoes, see Cell Shifts for Heptominoes.

## Arbitrary Solutions

 (1, 0)

 (1, 1)

### Trominoes

 (3, 0) (1, 0)

### Tetrominoes

 (4, 0) (2, 0) (1, 1) (2, 2) —

### Pentominoes

 (5, 0) (1, 0) (1, 0) (1, 0) (6, 0) Mike Reid (1, 0) (1, 0) (3, 0) (1, 0) (5, 0) (1, 0) Mike Reid —

### Hexominoes

 (6, 0) (1, 1) (2, 0) (3, 3) Mike Reid (3, 3) (2, 0) (1, 1) (12, 0) (1, 1) Mike Reid (1, 1) (2, 2) (1, 1) (2, 2) Mike Reid (3, 3) (6, 0) (4, 0) Mike Reid (4, 0) (1, 1) (2, 0) Mike Reid (3, 3) (1, 1) Mike Reid (2, 0) (1, 1) (1, 1) (6, 6) (1, 1) (4, 0) Erich Friedman — (1, 1) — (3, 3) (2, 0) (3, 3) (3, 3) (2, 2)

## Birotary Solutions

 (1, 0)

 (2, 0)

### Trominoes

 (3, 0) (1, 0)

### Tetrominoes

 (4, 0) (4, 0) (2, 0) (4, 4) —

### Pentominoes

 (5, 0) (1, 0) (1, 0) (1, 0) (6, 0) (1, 0) (1, 0) (3, 0) (1, 0) (5, 0) (1, 0) Mike Reid —

### Hexominoes

 (6, 0) (2, 0) (2, 2) (6, 0) (6, 0) (2, 0) (2, 0) (12, 0) (2, 0) (2, 0) (4, 0) (2, 0) (4, 0) (6, 0) (6, 6) (4, 0) (4, 0) (2, 0) (2, 2) (6, 0) (2, 0) (2, 2) (2, 0) (2, 0) (12, 0) (2, 0) (4, 0) Erich Friedman — (2, 0) — (6, 0) (2, 2) (6, 0) (6, 0) (4, 0)

Last revised 2013-07-01.

Back to Polyform Cell Shifting < Polyform Compatibility < Polyform Curiosities
Col. George Sicherman [ HOME | MAIL ]