This is a fairly tedious problem. The only way to solve that I see is to simply list out all the possible cases and go through each one by one. You'll use the triangle inequality theorem to see if the triangles can be formed. The theorem says that a+b > c must be true for all sets of pairs. In other words, take any two sides and add them up. The sum must be greater than the third side.
Going through all the combos possible (that I could find), this is what I got Blue9,Green7,Orange4 Blue9,Green7,Purple12 Blue9,Green7,Red3 Blue9,Green7,Yellow5 Blue9,Orange4,Purple12 Blue9,Purple12,Yellow5 Green7,Orange4,Yellow5 Green7,Red3,Yellow5 Orange4,Red3,Yellow5
In all, I count 9 cases. So the answer to problem 1 is 9.
For problem 2, the best way may be to pick two segments at a time instead of one. However, I have a feeling that will take just as long as the first method. I haven't tried it out. Even though going through the rods one at a time takes a while, it's probably the best option so you don't overlook any cases.
