Answer and Explanation for Solution: With a little bit of work, you can discoverthat only one solution is possible: the kids must be 1, 5, and 8 becausea solution with the same A and B involving twins (2, 2, and 10) is ruledout by the middle son with red hair. The product of both are 40 and the sumsare 14. The basis for that solution is as follows: The postman's firststatement tells us that x*y*z = A; the postman's second statement tellsus that x+y+z = B; and the third statement tells us that x does not equaly does not equal z. We know that the three ages are different because the"middle" son has red hair, which implies that none of thechildren are twins. The most important piece of information in the puzzleis that there must be at least two sets of numbers that add together to givethe same sum and multiply together to give the same product -- or hewould have been able to figure it out without the third parameter aboutthe middle child (no twins). Thus, there are two sets of threenumbers x1, y1, & z1 and x2, y2, & z2 where x1 + y1 + z1 = x2 + y2 + z2and x1 * y1 * z1 = x2 * y2 * z2 . It is not the fact that there are notwins, but the fact that the postman could NOT figure out the answer UNTILhe knew that there were no twins that is essential. He needed toknow that fact in order to solve the problem, and there must be only twodifferent sets of age choices for the kids (one with twins; one withou t twins)that produce equivalent A's and B's.
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