# Globular 3-4-5 Pentomino 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.

Issue 23 showed 3-4-5 dissections for pentominoes F, I, L, P, T, V, and Y. Some dissections were found by the editor, Rodolfo Kurchan. The others were found by Mike Reid.

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.

## Globular and Subglobular Dissections

Here I require that the pentomino 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 subglobular variants. If you find a solution of either kind with fewer pieces, please write.

For globular 3-4-5 dissections of smaller polyominoes, see Globular 3-4-5 Small Polyomino Dissections.

### 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 Pentominoes

Here are minimal known results for all 12 pentominoes. BZ = Berend van der Zwaag; ET = Edo Timmermans; GS = George Sicherman; GT = Gavin Theobald; HP = Helmut Postl; LZ = Livio Zucca; MR = Mike Reid; RR = Robert Reid.

5F5I
GS67MR45
5L5N
BZ55GS57
5P5T
LZ45GT67
5U5V
No solution.
GT6MR55
5W5X
GT78RR79
5Y5Z
No solution.
HP5ET6GT6

## Subglobular Dissections for Pentominoes

The unconditional dissections shown above for pentominoes F, I, L, P, T, and V are already subglobular.

5N5U5W5X5Y5Z
No solution. No solution.
6ET7ET86

Last revised 2024-07-17.

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