Intellectual power of a dragon pack?

Question

Dragons have to meet for a brainstorm in a convention center. The delegates have to be selected to provide the maximum efficiency of the brainstorm session. A dragon has any amount of heads, and for any N, any amount of N-headed dragons is available if needed. The problem is that the size of the convention center is limited so no more than 1000 heads can fit into the assembly hall. The intellectual power of a dragon pack is the product of head numbers of dragons in the pack. How should an optimum pack look like (the total number of dragons, the head number distribution)?

taken from
http://www.math.utah.edu/~cherk/puzzles.html

ermm...some explanation that goes with your answers would be appreciated.

Answer 1

Edit: Did you still want more details? If so, if you ask a specific question, I will try to answer it.

There should be 332 3-headed dragons and 2 2-headed dragons.

To see why, note that there is no point to having any single headed dragons, since that takes up one head and does not increase the brain power. Also note that there is no point to having a k-headed dragon for any k>3, since you can replace the k-headed dragon with a 2-headed dragon and a (k-2)-headed dragon and the brain power from those k heads goes from k to 2(k-2), and for k>3, 2(k-2) >= k.

So you know the most brain power is achieved with 2-headed and 3-headed dragons. Finally, notice that there is no point in having more than three 2-headed dragons, because you can replace any three 2-headed dragons with two 3-headed dragons and increase the brainpower from those 6 heads from 8 to 9. Thus the optimal collection of dragons is 332 3 headed and 2 2-headed dragons, and in general with h heads (where h >1), you should have 2*quotient(h/6) 3-headed dragons and remainder(h/6)/2 2-headed dragons.