Riddle: [QUOTE] Consider the isosceles triangle with base length, b = 16, and legs, L = 17. By using the Pythagorean theorem it can be seen that the height of the triangle, h = √(172 − 82) = 15, which is one less than the base length. With b = 272 and L = 305, we get h = 273, which is one more than the base length, and this is the second smallest isosceles triangle with the property that h = b ± 1. Find ∑ L for the twelve smallest isosceles triangles for which h = b ± 1 and b, L are positive integers. [/QUOTE] h = b ± 1. b/2 = h/2 ± 1/2. L = sqrt [ h^2 + (h/2 +/- 1/2)^2 ] = sqrt [ 5/4h^2 + 1/4 +/- h/2 ] a = h/2 L = sqrt[ 5a^2 + 1/4 ± a ] If $a$ is whole, then L^2 will have a residue of 1/4 which means L cannot be whole. Therefore $h$ is odd.