Determine the maximum value that is obtained by multiplying together a set of positive integers which are all different and whose sum is 70.

Basically you want the numbers to be as small as possible.

2+3+4+5+6+7+8+9+10+11 = 65 (note consecutive)
These numbers give the maximum product for a sum of 65.
Increase the 11 to a 12 and you have the numbers that give the maximum product for a sum of 66.
Increase the 10 to an 11 and the sum is 67.
Increase the 9 to a 10 and the sum is 68.
Increase the 8 to a 9 and the sum is 69.
Increase the 7 to an 8 and the sum is 70.

2+3+4+5+6+8+9+10+11+12 = 70 and the product is maximal.

This continues to
3+4+5+6+7+8+9+10+11+12 = 75
The trick is getting to the next level
3+4+5+6+7+8+9+10+11+13 = 76
2+3+4+5+6+7+8+9+10+11+12 = 77
then shift the numbers up one by one as before.

Posted by Jer on 2009-10-30 15:23:27