Pizza Hut Math (equal-sides sets of numbers with same sum and product) Pizza Hut, in honor of $\pi$ day, posted this problem:

Our school’s puzzle-club meets in one of the schoolrooms every Friday after school.
Last Friday, one of the members said, “I’ve hidden a list of numbers in this envelope that add up to the number of this room.” A girl said, “That’s obviously not enough information to determine the number of the room. If you told us the number of numbers in the envelope and their product, would that be enough to work them all out?”
He (after scribbling for some time): “No.” She (after scribbling for some more time): “well, at least I’ve worked out their product.”
What is the number of the school room we meet in?”

Is there a non-brute-force way of doing this problem? Additionally, I've found multiple same-sized sets of numbers with the same product and sum, so I imagine I am utterly misunderstanding the problem anyway (e.g. 2 triples that sum to 23 and multiply to 360, 2 triples that sum to 22 and multiply to 360, 2 triples that sum to 21 and multiply to 240, etc.). 
 A: Well, here is my solution (tl;dr my answer is 12)
First, we are looking for a number. We can write all possible unique "sum representations" of that number with different length. Every representation is a list of sorted numbers. For example 3 can be represented as [[3], [1 2], [1 1 1]]. Next, for every representation we can calculate product of the elements in it, for 3 it will be [3, 2, 1]. Also for every representation we can find number of elements in it: [1, 2, 3]. We in the "sum representations" of a number we want to find such representations that have same product of elements, but only one. I am sorry, I don't know how to explain it better (English is not my native). But let me explain why 23 is not an answer. 
Indeed, [4 9 10] and [5 6 12] are two sum-representations of 23 with product equals to 360. But, there is another representations of 23: [1 2 8 12] and [2 2 3 16] that give product 192. Thus, girl would not be able to guess the product in case of room number 23, because there is a lot of groups of representation (226) that will give the same product. 
Long story short, I was not able to find a number lower then 12 that would give me two representations with the same product. 12=1+3+4+4=2+2+2+6. 13 already have two different repeated products: 36 ([1 6 6] and [2 2 9]) and 48 ([1 1 3 4 4] and [1 2 2 2 6]). As you may notice, any other number bigger than 13 will have at least two representations with the product 36 and at least two with the product 48. So, 12 is the only solution. 
I am sorry for quite complicated explanation. I am not doing math on professional level, so I do not know all terminology of the number theory. I will be glad to answer any questions. 
A: I think it's worth stepping back a little bit and seeing exactly what's said, and therefore what's known. I think this problem may be deceptively involved.
First, the problem is worded poorly. We have no choice but to think the girl doesn't know the number of the room, as she says "That's obviously not enough information to determine the number of the room."
Second, the boy says that knowing the number of numbers and their product would not be enough to work them all out. By "them all," we must assume that the boy means not just the sum, but each individual number in the sum.
Third, from just knowing this, the girl is able to determine the product. This means two things. One, the room number in question must have a unique pair (I suppose there could be more than two) of sets of numbers that sum to the room number and have the same product. That is, the room number can't be 21 with 3 numbers listed in the envelope, for instance, because {1,8,12} and {2,3,16} both sum to 21 and multiply to 96, but {2,7,12} and {3,4,14} also sum to 21 but multiply to 168. Knowing that the number and the product aren't enough to determine all numbers on the list is not itself enough to uniquely determine the product.
The problem is that the girl has no reason to know, from what she's been told, what I just said in the paragraph above. If the room number were 21, the boy could tell her the same thing he already did tell her, and she still wouldn't be able to determine the numbers uniquely, nor would she be able to determine the product. I think there's some sloppy wording going on here that makes the problem unfortunately impossible to solve as stated.
A: Its important to capture two things here 1] number representation in sets, 2] product of those numbers. 
If you know the product and the corresponding number of numbers in the set then you can actually determine the Room number. For example lets say the product is 12 the set becomes [1, 12], [4,3] [2,2,3]. Now here we have two, 2 number sets both having different sum so can't predict the room number but given that there are three numbers we can predict the room number as 7
So, as you increase the product (from 12 to 13, 14, 15, 16 and so on) you may get different number of sets with different distinct sum and thats endless sequence.
The only possible sequence looks here is for product value 2 and the set as [1,2] (since we have more than one number as said in question) with the sum as 3 which seems to me as the room number as well.
A: To solve this involves making a number of assumptions.
Assumption 0: The characters are infinitely smart, and are not lying etc.
Assumption 1: The list of numbers are all positive integers. Otherwise the problem is clearly unsolvable. 12=0+0+-1+13+sqrt(12)-sqrt(12) etc. Every room number could be made with a sum of an infinite list giving any desired product.
Assumption 2: The girl can now figure out the room number.
It follows that: The school has a finite set of class rooms and both the girl and boy know this.  This is the only way that the girl can rule out room number say, 13.  Because room number 13 doesn't exist in their school (or is otherwise inaccessible but the reason it is ruled out is outside the scope of this problem!).  If room 13 did exist, she would not be able to say “well, at least I’ve worked out their product.” based on the boy's answer of "no".
The only way we as a reader can know that 13 is discounted is because we must assume that room 13 does not exist in their school and the girl knew this.  That's the piece of information that the girl knows but we, as a reader, do not and must infer.
We must discount the existence of any room number that has more than one duplicate product, such as room number 13 with 2 such duplicate products: 36 and 48.
36 ([1 6 6] and [2 2 9]) and 48 ([1 1 3 4 4] and [1 2 2 2 6])
Any room number greater than 12 will have at least the products 36 and 48 from 2 different sets of the same size (by tacking on 1s to the list).
12 is the only room number with exactly 1 duplicate product (48), that has multiple sum representations of the same size (1+3+4+4=2+2+2+6). Any room number less than 12 would be ruled out because the boy's answer would have been "yes".
The author of this problem is none other than John H. Conway, so I was resistant to accept that this is just a poorly worded problem with no solution.  I'm very satisfied to realize that the problem is solvable and requires very subtle assumptions. 12 is the correct room number, but we also learnt that 12 is the highest room number in the school.
