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Sticks and Triangles

Age 7 to 11 Challenge Level:

I made some triangles out of sticks. The sticks are all the same length.

I can make a triangle out of three sticks:

triangle from 3 sticks

What about four sticks? Five sticks? Six sticks?

I can make two different triangles out of ten sticks:

two triangles from 10 sticks

Can you find any others?

What about triangles made from twenty sticks?

What about all the lengths in between?

Can you make any rules about the numbers of sticks that make the most triangles?