What a nice and curious problem. It's rather startling at first that the answers needn't be integers.
*I don't even know that 5/3 is the best answer for the example above,*
It seems most implausible that it isn't. Hmm, yes, let's see: in a better solution, each 5 stick must be cut into at most 2 pieces, obviously. The resulting (wlog) 14 pieces must be assembled into five 7-sticks so one 7-stick must get at most two pieces. One of these must be at least 3.5 long. Whichever 5-stick that came from, you had a length of 1.5 left over, which is too short. |