The children are 1, 5, and 8. Given that Holmes couldn’t deduce their ages despite knowing the sum of their ages, I could have worked out that the sum of their ages was 14. This is because from all the possible combinations that would yield the product of 40, only two have the same sum: 1, 5, 8, and 2, 2, 10. The final clue was necessary to choose between these two options. If two of the children were 2, then they would have to be twins. The final clue rules out this possibility.