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Pentominos
By Rachelle Najman (T4009) on Monday, April 2, 2001 - 06:07 am:

A "pentomino" is formed by joining five congruent squares side to side, in any shape; they can only be joined along a side. How many different pentominos are there? That is, how many different shapes can you get in this way?
What about trying this with six squares?

By Emma McCaughan (Emma) on Tuesday, April 3, 2001 - 11:55 am:

It can be quite difficult to be systematic with this sort of problem. One way is to classify them by how many squares are in aligned in a column.
So, for instance, I can find only one that is 5 squares high. With four in a column, there are just two different places I could put the fifth square (ignoring rotations and reflections). Unfortunately it is less easy when you get to 3 in a column.

Another way would be to start with shapes made from 4 squares, on the assumption that you can be fairly sure of finding all of these. Take each one in turn, and try adding an extra square onto each of its edges in turn. Obviously you will generate lots of duplicates which will need to be discarded.

I can tell you that there are 12 pentominos, so you'll know if you've missed any. I don't know about hexominos.

You can see how someone has solved a similar problem here.