Pentominoes were invented by Solomon Golomb for a presentation at a national junior high school mathematics competition, May 10, 1996. That's not very long ago! Since then, mathematics enthusiasts of all ages have been playing with them.

Solomon Golomb is well known in mathematical circles as an inventor of interesting bits of recreational mathematics, including Rep-tiles which are shapes that can be dissected into identical smaller copies of the original shape. He also invented Golomb's Ruler. This, he claims, is a ruler which has marks only at 1, 3, 6 and 7 centimetres (or inches in his imperial units) yet it can be used to measure every integer distance from 1 to 6 centimetres in length.

Pentominoes are figures which are made up of joining 5 squares together. The squares are all the same size.



Professor Golomb

There are 12 pentominoes. None are the same as the mirror image or any way of turning any other pentomino. They are all quite different. Above are two of them. All you need to do is find the other 10.

Can you find all 12 pentominoes?

Once you have all 12 of them you can combine them to make shapes. Some or all of them can be used.

Cut out the pentominoes so you can move them into these shapes. You have only these 12 pieces to use - but you can use the same pentomino twice in any shape.

Start with a 10 x 3 rectangle - and fill it with pentominoes.

Now fill the the 5 x 5 square with pentominoes.

Can you construct the following rectangles which use all twelve pentominoes?
15 x 4 little squares
12 x 5 little squares
10 x 6 little squares

Of course then there are trominoes (3 little squares joined into shapes), tetrominoes (4 little squares joined into shapes) and hexominoes (6 little squares joined into shapes).
How many of each of these you can find?