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Message: <blockquote data-quote="Iron Sheep" data-source="post: 1344307" data-attributes="member: 4965">I hate these sorts of weird number fact problems, and I'm a mathematician!  That probably explains why I'm not a number theorist, though.  Analysing what the problem says:[SPOILER]B's first comment means that neither X nor Y is prime, otherwise B could simply factor and say the answer.  So at least one of the numbers must be a composite number.A's response means that no matter how you split the sum (X+Y) as a sum of two numbers A and B greater than 1 and less than 100 (so X+Y = A+B), you never have both numbers A and B prime at the same time, since then A couldn't already be sure that B didn't know the answer.  For example, the sum couldn't be 12, since 5+7=12, and 5 and 7 are prime. However 11 does have this property, since 11 = 2+9 = 3+8 = 4+7 = 5+6.  Call this property of a number (P).B's second comment means that of all the possible ways of factoring the product only one of them has a sum with property (P).A's second response means that of all the ways to pick A and B to give the same sum as (X+Y), only one of them has just one way of factoring to give a sum with property (P).[/SPOILER]Having determined all that, it's just a matter of going through and exhausting all the possibilities until you find the right one.  I threw my computer at the problem, since I didn't feel like using my brain too much, and got the following:[SPOILER]The numbers 4 and 13 are a solution.  The number A is given is 17, and the number B is given is 52.A knows that B can't immediately tell what the factors are, since 17 = 2+15 = 3+14 = 4+13 = 5+12 = 6+11 = 7+10 = 8+9, and none of those pairs of numbers are both prime.The factorizations of 52 are 2*26 and 4*13.  2+26 = 28, but 28=11+17, which are prime, so A wouldn't have said what he did if the sum was 28.Looking at the other possibilities of products from A's information we have:2*15 = 30, with factors 2*15, 3*10, 5*6.  Both 17 (=2+15) and 11 (=5+6) have property (P), so if the product was 30, B wouldn't have known which was right.  So A knows the product isn't 30.3*14 = 42, with factors 2*21, 3*14, 6*7.  Both 23 (=2+23) and 17 (=4+13) have property (P), so A knows that the product isn't 42.5*12 = 60, with factors 2*30, 3*20, 4*15, 5*12, 6*10. Both 23 (=3+20) and 17 (=4+13) have property (P), so A knows that the product isn't 60.6*11 = 66, with factors 2*33, 3*22, 6*11. Both 35 (=2+33) and 17 (=4+13) have property (P), so A knows that the product isn't 66.7*10 = 70, with factors 2*35, 5*14, 7*10. Both 37 (=2+35) and 17 (=4+13) have property (P), so A knows that the product isn't 70.8*9 = 72, with factors 2*36, 3*24, 4*18, 6*12, 8*9. Both 27 (=3+24) and 17 (=4+13) have property (P), so A knows that the product isn't 72.So the only way that B could have known the factors is if the product was 52, so A knows that the numbers are 4 and 13.[/SPOILER]I think this one's a bit too tough (mainly because of the size of the solution space you need to search) to throw at your players, unless your players are members of a math olympiad team.Of course there's probably some neat way of seeing the answer quickly.Corran</blockquote>

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