OPTION B: 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?

--it belatedly occurs to me, that by "list" of numbers, Conway likely did not mean "set," he meant "multiset." If so, my alleged solution might be blown out of the water. I must say, though, that if Conway intended my solution as a decoy solution, which cleverly had (what seems to be a unique) answer, reassuring me of its "correctness" and "wondrousness"... well... then that would make Conway even more fiendish!! Speaking of my answer "19" being unique, we have these two lines nn=3 pr=90 su=20: 20=1+9+10, 90=1*9*10, 20=2+3+15, 90=2*3*15 nn=4 pr=144 su=20: 20=1+2+8+9, 144=1*2*8*9, 20=1+3+4+12, 144=1*3*4*12 ruling out the answer 20 since it does not uniquely determine a product. But now, increase 20 to 21 by adding 1 to every sum, leaving every product unaltered. Keep adding 1. In that way, we see the multiset version of the problem cannot have any answer above 19. So the answer must lie within the 20 choices 0,1,2,...,19.