The numbers 1-10 are to be entered into the 10 boxes shown below, so that each number is used exactly once:
\[P = (\square + \square + \square)(\square + \square + \square + \square + \square + \square + \square).\]
What is the maximum value of P? What is the minimum value of P?
To maximize the number, we want the numbers to be as close in value as possible. If we take any number as the sum of the two factors, say 12, we can see that 6*6 is larger than 11*1.
We know that by adding the numbers from 1-10 we get 55, so we want values of 27 and 28.
The first thing that comes to mind would be 10+9+8*7+6+5+4+3+2+1, which is 27 and 28 when multiplied, is 756.
To minimize the amount, the numbers have to as far apart as possible.
To get this, we use the smallest numbers, 1,2, and 3 for the 3 first boxes, and the rest of the numbers in the other boxes.
1+2+3 is 6, and 55-6 is 49. 6*49 is 294.
So the largest number would be 756, and the smallest would be 294.