1. In any triangle, if a and b are the lengths of the shorter sides and c is the length of the longest side, then a + b > c.
So, for a triangle to have its longest side with length c = 7, the two shorter sides can have lengths
• a = 2 and b = 6 (2 + 6 = 8 > 7)
• a = 3 and b = 5 or b = 6
• a = 4 and b = 5 or b = 6
• a = 5 and b = 6
which means there are 6 ways for Joe to build such a frame.
2. In general, the number of frames Joe can make with longest side c is a matter of finding the number of positive integer solutions to
a + b > c
such that a < b < c.
c must clearly be greater than 3, because the smallest values for a and b are 1 and 2, and 1 + 2 = 3. For all other choices of c, we can set a = 2 and b = c - 1, or a = 3 and b = c - 2, or a = 4 and b = c - 3, …
