products of sums
The list of positive integers that add up to 1 is (1), and the product of this list (if you allow unary products) is also 1.
There are two lists of positive integers that add up to 2, and they yield two different products: (2) (with product 2), and (1,1) (with product 1).
There are several lists of positive integers that add up to 3, and they yield several different products: (3) (with product 3), (2,1) (with product 2), and (1,1,1) (with product 1).
If n is a positive integer, what is the maximum product that can be formed of a list of positive integers that sum to n?
Solution:
The rule is to get as much 3 as possible. But if separating the last three from the total sum leave a 1, the last 3 will not be separate, and will separate the last 4 integer together as 2 + 2. If separate the last three leave a 2, it will be fine because 2 * 3 is the biggest product of sum 5.
Finlay, the common solution of this problem is: divide the sum by 3. Name the integer N. And if the reminder is 1, the power N minus 1, and product equals 3^(N - 1) * 2 * 2. If the reminder is 2, product equals 3^N * 2.