The first question you need to ask is, for what sums S can Sam say to Max "You don't know what X & Y are"? How about S=10? Well, if S=10, we could have... X=3,Y=7... => P=21. But if P=21, Max knows X=3,Y=7. So Sam can't make the claim he did, because X & Y MIGHT be (3,7). On the other hand, if S=13... X=3,Y=10... => P=30 => could have (5,6) or (3,10)... Max doesn't know or X=4,Y=9... => P=36 => could have (6,6) or (4,9)... Max doesn't know or X=5,Y=8... => P=40 => could have (4,10) or (5,8)... Max doesn't know or X=6,Y=7... => P=42 => could have (3,14) or (6,7)... Max doesn't know Regardless, if S=13, Sam can say with certainty that Max doesn't know X & Y based on P. Of course, there are other values of S for which this is true. This is how figuring out the problem STARTS (there's still a lot of work ahead ;-) Hope this helps.