We consider one dimensional disconnected polyominoes, collections of lattice squares in a line. We ask which can tile rectangles, and how many polyominoes are needed. The smallest number of polyominoes that tiles some rectangle is called the order of the polyomino.

Some of these are easy, as they will tile an n×1 rectangle for some n. For example, the polyomino tiles a 6×1 rectangle. Others are more difficult, like , which tiles a 15×12 rectangle. Others, like , cannot tile a rectangle at all.

So which polyominoes can tile a rectangle? Of these, what is the order?

I wondered whether increasing the rectangle to 3 (or more!) dimensions would help polyominoes tile. I conjectured that all polyominoes tile some d-dimensional cube. Reid came up with this counterexample, which can obviously never tile the corner of any cube:

Patrick Hamlyn wrote a computer program that found many minimal tilings, and the number of tilings for various sizes of rectangles.

Joe DeVincentis found some minimal tilings, and he did all his tilings by hand! He notes that in every minimal tiling, the longer dimension is divisible by the number of squares in the polyomino. He asks whether this is true in general. In fact, he challenges anyone to find a tiling of a 1-dimensional polyomino in which the area divides neither dimension of the tiled rectangle. Reid has managed to show that such a polyomino must have area at least 6.

Aaron Meyerowitz has written several papers on tiling disconnected polyominoes in 1 dimension. He told me that Andrew Adler and F. C. Holroyd proved in 1981 that any 1 dimensional triomino tiles some nx1 rectangle. This was also known to Klarner much earlier. He also told me that it follows from one of his papers that the order of such a triomino is the 1 dimensional order. It is not known whether this is true for other polyominoes which tile in 1 dimension.

Meyerowitz also told me that Don Coppersmith proved in 1985 that every tetromino (not necessarily 1 dimensional) tiles the plane. There is a hexomino that does not tile the plane: . It is not known whether every pentomino tiles the plane.

Meyerowitz also asks 2 questions: "If a polyomino tiles a half strip, must it tile some rectangle?" and "If a polyomino tiles the half line, must it tile a finite interval?"

Rick Eason sent me a detailed description of the packable boxes of .

Here are the results for small length polyominoes:

Polyomino	Order	Dimensions	Finder	Prover
	2	4x1
	2	6x1
	3	6x1
	2	8x1
	none			Juha Saukkola
	2	6x1
	2	6x1
	4	8x1

Polyomino	Order	Dimensions	Finder	Prover
	2	10x1
	24	20x6	Erich Friedman	Patrick Hamlyn
	24	16x6	Joe Devincentis	Patrick Hamlyn
	2	8x1
	21	12x7	Erich Friedman	Erich Friedman
	2	8x1
	3	9x1
	6	18x1
	5	10x1

Polyomino	Order	Dimensions	Finder	Prover
	2	12x1
	none			Mike Reid
	none			Erich Friedman
	36	15x12	Patrick Hamlyn	Patrick Hamlyn
	2	10x1
	36	15x12	Juha Saukkola	Patrick Hamlyn
	none			Erich Friedman
	2	10x1
	none			Erich Friedman
	2	8x1
	2	8x1
	2	8x1
	60	24x10	Patrick Hamlyn	Patrick Hamlyn
	2	8x1
	none			Mike Reid
	4	12x1
	4	12x1
	3	9x1
	6	12x1

Polyomino	Order	Dimensions	Finder	Prover
	2	14x1
	none			Mike Reid
	none			Mike Reid
	70	30x14	Mike, Patrick	Patrick Hamlyn
	2	12x1
	none			Mike Reid
	32	24x8	Mike, Erich	Patrick Hamlyn
	none			Erich Friedman
	2	12x1
	none			Mike Reid
	none			Erich Friedman
	none			Erich Friedman
	48	30x8	Mike, Joe	Patrick Hamlyn
	78	30x13	Mike, Patrick	Patrick Hamlyn
	2	10x1
	90	50x9	Mike, Joe	Patrick Hamlyn
	none			Mike Reid
	2	10x1
	45	25x9	Mike, Joe	Patrick Hamlyn
	96	60x8	Mike, Patrick	Patrick Hamlyn
	2	10x1
	2	10x1
	3	12x1
	84	28x12	Mike, Patrick	Patrick Hamlyn
	8	32x1
	66	24x11	Mike, Patrick	Patrick Hamlyn
	3	12x1
	240?	20x48	Mike Reid
	6	24x1
	50	20x10	Mike, Joe	Patrick Hamlyn
	102	24x17	Mike, Patrick	Patrick Hamlyn
	4	12x1
	10	30x1
	12	36x1
	7	14x1

Polyomino	Order	Dimensions	Finder	Prover
	2	16x1
	none			Mike Reid
	none			Mike Reid
	none			Mike Reid
	54	42x9	Patrick Hamlyn	Patrick Hamlyn
	2	14x1
	none			Mike Reid
	36	28x9	Patrick Hamlyn	Patrick Hamlyn
	none			Patrick Hamlyn
	none			Erich Friedman
	2	14x1
	none			Mike Reid
	none			Mike Reid
	none			Mike Reid
	none			Mike Reid
	none			Mike Reid
	?
	?
	2	12x1
	30	18x10	Patrick Hamlyn	Patrick Hamlyn
	none			Mike Reid
	none			Mike Reid
	2	12x1
	none			Mike Reid
	180?	108x10	Mike, Patrick
	none			Mike Reid
	none			Mike Reid
	none			Mike Reid
	36	24x9	Mike, Patrick	Patrick Hamlyn
	?
	none			Mike Reid
	none			Mike Reid
	2	12x1
	none			Mike Reid
	2	12x1
	2	10x1
	2	10x1
	?
	2	10x1
	156?	60x13	Mike Reid
	2	10x1
	2	10x1
	2	10x1
	234?	90x13	Mike Reid
	none			Mike Reid
	72	30x12	Mike, Patrick	Mike Reid
	108	45x12	Mike, Patrick	Patrick Hamlyn
	2	10x1
	240?	100x12	Mike Reid
	2	10x1
	?
	48	20x12	Patrick Hamlyn	Patrick Hamlyn
	none			Mike Reid
	?
	4	16x1
	88	32x11	Mike Reid	Patrick Hamlyn
	4	16x1
	48	16x12	Patrick Hamlyn	Patrick Hamlyn
	120?	40x12	Patrick Hamlyn
	?
	60	24x10	Mike Reid	Mike Reid
	4	16x1
	6	24x1
	none			Mike Reid
	60	20x12	Patrick Hamlyn	Patrick Hamlyn
	?
	5	15x1
	4	12x1
	4	12x1
	4	12x1
	8	16x1

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/3/99.