Kobon Fujimura posed the problem (fairly recently) of finding the maximum numbers of triangles (not overlapping) for a given number of straight lines.
Students could be asked to explore this, for up to 10 lines:
n = 3 , 5 and 7 are shown, giving totals of
1 , 5 and 11 respectively
for n = 4 , 6 and 8 the totals are 2 , 7 and 16
can students produce diagrams for these?
beyond small numbers of lines the diagrams become complex, with tiny triangles
for solutions see e.g. Wolfram
n = 18
n = 20
Saburo Tamura proved that an upper limit to the number of triangles is given by the largest integer less than or equal to
n(n - 2)/3