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I was exploring a puzzle in which headless match sticks had to be moved to make a different number of triangles.

I made one small triangle

Triangle from three matches. $3$ matches

I made it into $4$ small triangles by adding $6$ matches.
Four triangles. $9$ matches

I added another row and counted the number of small triangles and counted the matches.

9 triangles from matches.


I made a table of my results and continued adding rows. I found many patterns.

Have a go and see what patterns you can find. You do not have to use match sticks (or cocktail sticks) - drawing lines will do just as well.

Find a good way to record your results. See if you can predict the numbers for rows of triangles you have not drawn.

When you have done all you can with triangles, see if you get the same sort of results with squares. Then think of other shapes which might make number patterns as they grow.