Before I give my own reasoning, I am not fully convinced by the reasons you gave. How did you narrow down your "unique" solution to only three possible choices. You did not give any reason! In other words, what makes them unique?? Also, when you eliminated your last answer (of the remaining two) saying 18 appears above, then 2 also appears above. I don't get your reasoning. -------------------------- Here's what I would do: To begin with, there are 11 possible combinations. 1+2+36=39 1+3+24=28 1+4+18=23 2+2+18=22 1+6+12=19 1+8+9=18 2+3+12=17 2+4+9=15 2+6+6=14 3+3+8=14 3+4+6=13 In all of the cases above, the product is 72, but their sums are different except for 2, 6, 6 and 3, 3, and 8. The number of windows the mathematician sees is 14. That's why he needs one more hint to know which one among the two is the answer. The number of windows has to be 14 because, if there was another number, say 18, the mathematician would know the answer right away, and it would be: 1, 8, and 9. So, because the sum of their ages (the number of windows) was 14, the mathematician asks for one more hint to which the housewife says: My eldest son has red hair. If the eldest son has a red hair, then the answer cannot be 2, 6, 6 because then you would have twins there. So, the answer must be 3, 3, 8. --------------