I can join two equilateral triangles together along their edges like this:

How many different ways are there to join three equilateral triangles together? (You must match an edge to an edge.)

How do you know you have got them all? You may find it useful to print out and cut up triangles from this sheet.

Have you got a system for checking that you have them all?

For a further challenge, you might like to look at Tri-Five.