Let's start with an easy problem : What three positive integers have a sum equal to their product?

answer: (1,2,3), of course.

This puzzle can easily be transformed into a D4 problem:

For what values of k will the question "**What k positive integers have a sum equal to their product?**" have only one unique set of integers for an answer?

Clearly for k=2 the answer is unique: (2,2) and so it is for k=4: (1,1,2,4).

List all other values of k below 1000.

(m,2,2,1,...1). The product = 4m so the sum = product if there are 4m - m - 4 ones, in which case there are 3m - 1 terms. So this works for k = 3m-1 where m>2, i.e. {5, 8, 11, ...).

We have already ruled out k = 3m-2 where m>=4, so above 12, k can only be a multiple of 3.

And above 4, k can only be a multiple of 2, so starting with 12 k can only be a multiple of 6

This leaves (below 60): 2,3,4,6,12,18,24,30,42,48,54

More elimination required