What is the highest product of numbers that sum to 10?

 I'm always on the lookout for maths puzzles, especially ones which can be described simply but require some work to find the solution.  The most popular article on my blog is the one "Can you use the digits 1,2,3 and 4 to make the numbers from 1 to 50."  It's depressingly popular, when I have spent so much more time on many of the other posts here.  Ah, well, that's life.

A puzzle I found recently goes like this: "What is the highest product (i.e. multiply them all together) of numbers that sum to 10?"

Simple enough. 

I started with the pairs, 1+9, 2+8 and so on:

1 * 9 = 9
2 * 8 = 16
3 * 7 = 21
4 * 6 = 24
5 * 5 = 25

Well, there we are.  The answer so far is 25.  The highest value we can reach for x (10 - x) is when x = 5 and 5 * 5 = 25.  We can even prove this, using differention:

If y = x(10-x) = 10x - x^2  

then dy/dx = 10 - 2x 

And the maximum value is when dy/dx = 0 and if 10 - 2x = 0 then x = 5.  QED

However, there are multiple other ways of making 10 by summing numbers together; nobody said it had to be just two numbers.

For example, 4 + 4 + 2 = 10, and 4 * 4 * 2 = 32, which is higher than the 25 we achieved previously.

A few more examples:

2 * 4 * 2 * 2 = 32
2 * 3 * 5 = 30
3 * 3 * 4 = 36

The highest that can be achieved, it seems, is 36.

Let's try another number - let's try summing to 15, and see if we can find a pattern.

3 * 6 * 6 = 108
5 * 5 * 5 = 125
3 * 3 * 3 * 4 * 2 = 162
3 * 4 * 4 * 4 * 4 = 192
3 * 3 * 3 * 3 * 3 = 243

Summing to 15, the highest product we can achieve is 243.

Summing to 19 (if you fancy a challenge) is 972, where 3 * 3 * 3 * 3 * 3 * 4 = 972.

Proof (or general principle)

When forming the product with the highest value, the highest value can be achieved by using 2s and 3s.

Every larger number can be broken down to 2s and 3s which will multiply together to form a larger product.

For example, 9 can be be broken down to 5 and 4, which will multiply to form 20 (which is higher than 9).

9 --> 5 * 4 = 20

5 -->  3 * 2 = 6

9 --> 3 * 2 * 4 = 24

4 --> 2 * 2 = 4 (hence it does not matter if we choose a 4 or two 2s).

7 --> 3 * 4 = 12

6 --> 3 * 3 = 9

Any number n can be broken down to 2s and 3s which will multiply together to form a product which is greater than n.  Hence, the largest product we can obtain for 10 is 3 * 3 * 3 * 2 = 18, and the general principle is to break down the sum number into 3s (and 2s if the number is not a product of 3) to obtain the highest product.