Cell Shifts for Polyominoes

Introduction

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
• Birotary Solutions

Arbitrary Solutions

Monomino

(1, 0)

Domino

(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
	—

Symmetric Variants

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

Monomino

(1, 0)

Domino

(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.