Kobon triangles

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:

some diagrams for
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

 (there could be 26 for 10 lines)



some examples:

n = 9 giving 21 triangles

n = 10 (shown symmetrically) giving 25 triangles (it might be possible to produce 26)

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

art work