Tuesday, 29 April 2025

Maths: Some shape puzzles

 Pentominos

Below, you can see the Pentominos (the last I heard, the name was a registered trade mark of Solomon Golomb). There are twelve of them, and they are each made up of five squares. You can lay them down either side up, and you can use them to make a variety of larger shapes.

Solomon Golomb’s Pentominos.

First, get some cardboard or light board of some sort, and make up a set, using the patterns you see here. Once you have done that, see if you can arrange them into a square, 8 x 8, with a small empty square, 2 x 2, in the middle somewhere.

Then try to see if you can arrange all of the Pentominos to make a 6 x 10 rectangle, then a 5 x 12, a 4 x 15 and a 3 x 20 rectangle. Golomb has published at least one book on the subject—try your local library to see if you can track it down. You can also make a 5 x 13 rectangle that leaves a hole in the shape of one of the Pentominos, while other challenges include making scaled-up versions of some of the Pentominos, using nine of the other pieces to make a model three times as long and three times as wide.

Another interesting challenge is the “double double” where you make the same shape with two Pentominos, make a second copy with two more Pentominos, and then make a double-sized model with the remaining eight Pentominos.

Why not make a set to use at home?

The Soma shapes

The Soma shapes are as shown below: they were the invention of Piet Hein, who pops up in other places in this book. There are six shapes made of four cubes, and one shape made of three cubes.

Piet Hein’s Soma shapes.

Use cardboard to make a set of templates that can be folded and glued into the shapes, or make a set from wooden blocks, or sew them from cloth and stuff them like cushions (this works if the basic cube is about 20 cm or more on a side), and then see if you can solve the puzzle.

The diabolic cube

The diabolic cube pieces.

This 19th century puzzle requires you to take the pieces you see here, and assemble them into a cube. Just make them up in the same way that you would make up the Soma pieces, and then get to work to fit them together. What could be easier?

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