Globular 3-4-5 Small Polyomino Dissections

History

Issue 21 of Puzzle Fun, published in August 1999, showed Robert Reid's dissection of an X pentomino at scale 5 into two X pentominoes, one at scale 3 and one at scale 4. Problem 609 was to find the same kind of dissections for the other 11 pentominoes, using as few pieces as possible.

Erich Friedman's Math Magic for December 2009 posed this problem, considering pentominoes and smaller polyominoes as well. He posted dissections by himself, Berend van der Zwaag, Mike Reid, Gavin Theobald, and me. He later posted improved dissections by Livio Zucca and Helmut Postl.

Here I show only dissections of the monomino, domino, trominoes, and tetrominoes. For dissections of pentominoes, see Globular 3-4-5 Pentomino Dissections.

Globular and Subglobular Dissections

Here I require that the polyomino at scale 4 be undivided. I call such a dissection globular. Globular dissections usually require more pieces than unconditional dissections.

Edo Timmermans has also investigated dissections in which the polyomino at scale 3 is undivided. For want of a prettier term, I call such dissections subglobular.

The first table below shows both unconditional dissections and globular ones. The unconditional dissections are taken from Math Magic.

The second table shows unconditional and subglobular dissections. If you find a solution of either kind with fewer pieces, please write.

In these tables, where Edo and I found dissections with the same number of pieces, I give Edo's solution. Such cases are tagged ET (GS).

Globular Dissections of Rectangles

A globular dissection of the monomino requires only 4 pieces. This dissection uses 5:

Because no pieces are rotated, the dissection can be stretched horizontally and vertically to form a rectangle with any dimensions. In particular, it provides a globular dissection of any straight polyomino. If the straight polyomino has 3 or more cells, this globular dissection is minimal.

Globular Dissections of Polyominoes

Here are minimal known results for all 12 pentominoes. BZ = Berend van der Zwaag; EF = Erich Friedman; ET = Edo Timmermans; GS = George Sicherman.

1O 2I 3I

EF 4 GS 4 EF 4 GS 4 EF 4 GS 5

3L 4I 4L

BZ 4 EF 4 GS 5 BZ 4 ET (GS) 5

4N 4T

GS 5 ET (GS) 7 GS 5 GS 6

1O	2I	3I

EF	4	GS	4	EF	4	GS	4	EF	4	GS	5

3L	4I	4L

BZ	4	EF	4	GS	5	BZ	4	ET (GS)	5

4N	4T

GS	5	ET (GS)	7	GS	5	GS	6

Subglobular Dissections for Small Polyominoes

The unconditional dissections shown above for the monomino, domino, straight tromino, and straight tetromino are already subglobular.

3L 4L

BZ 4 GS 4 BZ 4 ET 5

4N 4T

GS 5 ET 6 GS 5 ET (GS) 6

3L	4L

BZ	4	GS	4	BZ	4	ET	5

4N	4T

GS	5	ET	6	GS	5	ET (GS)	6

Last revised 2024-07-17.

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Col. George Sicherman [ HOME | MAIL ]