A Cliche Test Problem

Let's start by eliminating the strictly dominated strategies:

• For any number $n \ge 1$ , $1(n-1)<n$
• For any number $n \ge 5$ , $2(n-2)>n$ and $3(n-3) >n$
• For $n =4$ , $2 \times 2 = 4$ and $1 \times 3 < 4$ , so $2+2$ and 4 can be used interchangeably.

So to maximize the product of the partition that adds to 25, each of the integer will have to be either 2, 3 or 4

• If most of them are 2's you can have $2+2+2+2+2+2+2+2+2+2+2+3 = 25$ and the product of them amounts to $2^{11} \times 3 = 6144$
• If most of them are 3's you can have $3+3+3+3+3+3+3+4 = 25$ and the product of them amounts to $3^{7} \times 4 = 8748$
• If most of them are 4's you can have $4+4+4+4+4+2+3 = 25$ and the product of them amounts to $4^{5}\times 2 \times 3 = 6144$
• If half of them are 3's and half of them are 2's you have $2+2+2+2+2+3+3+3+3+3 = 25$ and the product of them amounts to $2^{5} \times 3^{5} = 7776$

As such the answer is $\boxed{8748}$

Mathematica

Max[Times@@@IntegerPartitions@25]

returns 8748

the numbers are

{4, 3, 3, 3, 3, 3, 3, 3} OR {3, 3, 3, 3, 3, 3, 3, 2, 2}