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

two triangles

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?

How many different ways are there to join four equilateral triangles together?
Have you got a system for checking that you have them all?

You might like to use this interactivity to try out your ideas. Click the red triangle to make a new triangle appear. You can move the triangles around by dragging with your mouse. If you hover your mouse over a corner, you will see a white circle where you can click to rotate the triangle.

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