Maximum size of Huffman codes for an alphabet containing 256 letters
I don't see how to approach the problem.


OK, here is how you can approach the problem.  Can you solve this problem if you replace the number 256 with the number 3?  How about 4?  5?  Try solving those special cases, then see if you spot a pattern...

This is a useful, general pattern.  When a problem is too hard, try to find a simpler version of it and think about that.  When you see a constant in the problem, try solving simpler versions where the constant is smaller (so the problem becomes easier to think about), then look for a pattern, try to generalize, form a hypothesis, and see if you can prove your hypothesis.