Age of Three Daughters

A census taker approaches a woman leaning on her gate and asks about her children. She says, “I have three children and the product of their ages is 72. The sum of their ages is the number on this gate.” The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, “I have to see to my eldest child who is in bed with measles.” The census taker departs, satisfied

For 72:

The prime factors of 72 are 2, 2, 2, 3, 3; in other words, 2 × 2 × 2 × 3 × 3 = 72
This gives the following triplets of possible solutions;
Age oneAge twoAge threeTotal (Sum)
Because the census taker said that knowing the total (from the number on the gate) did not help, we know that knowing the sum of the ages does not give a definitive answer; thus, there must be more than one solution with the same total.
Only two sets of possible ages add up to the same totals:
A. 2 + 6 + 6 = 14
B. 3 + 3 + 8 = 14
In case ‘A’, there is no ‘eldest child’ – two children are aged six (although one could be a few minutes or around 9 to 12 months older and they still both be 6). Therefore, when told that one child is the eldest, the census-taker concludes that the correct solution is ‘B’.