by Bill Graham

Before working too hard on this problem, enumerate the possible ages of the children and hope for some insight. How many ways can three numbers have a product of 36? Because the question involves a sum of the ages, include that too.

1×1×36	38
1×2×18	21
1×3×12	16
1×4×9	14
1×6×6	13
2×2×9	13
2×3×6	11
3×3×4	10

You'll notice one peculiarity of this list. All of the sums except for one are unique. Henrietta knew the value of the sum, yet did not know the ages of the children. Only one possiblity could cause this problem -- the number of the house across the street is 12, and the sum of the ages of the children is 13.

By simply revealing that there is an oldest child, the man allowed Henrietta to eliminate the 1-6-6 case that would have involved twins. (We'll ignore the possibilty that two six-year olds might be distinguished by precise moment of birth because this is a math problem.)

So the ages of the children are 2, 2, and 9.

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