# project-euler / project-euler / 138 / 138-analysis.txt

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```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. a = n + 1/2 L^2 = 5(n+1/2)^2 + 1/4 ± (n + 1/2) = 5n^2 + 5n + 5/4 + 1/4 ± (n + 1/2) = 5n^2 + 5n + 3/2 ± (n + 1/2) Two sub-cases: 1. L^2 = 5n^2 + 6n + 2 = 2. L^2 = 5n^2 + 4n + 1 = ```