systematic counting of triangles

more complicated to keep track of than 'rectangles in a rectangle', this fairly well known puzzle:

to count the total number of triangles in this shape (usually with three lines from each lower vertex rather than the two shown here)

has been cleverly analysed by Ethan Siegel

and has an easy to recognise generalisation

students can think about ways to systematically count all the options

[in his blog article, Ethan highlights several incorrect methods that he has found]












Ethan proposes considering each node in turn, up from the bottom one so that all new triangles are counted and none are counted twice:

1 + 2 + 3
2 + 3 + 4
3 + 4 + 5










with differing numbers of additional lines, clear patterns can emerge:

David Wells, also cleverly, shows how this problem relates to that of counting the number of rectangles in a rectangle: