From (i), it is clear that A has at least 3 children and the number of children A has form the following sequence:
3, 6, 9, 12, 15, 18, 21, 24, ……..
From (ii), it is clear that B has at least 4 children and the number of children B has form the following sequence:
4, 8, 12, 16, 20, 24, ……..
From (iii), it is clear that C has at least 5 children and the number of children C has form the following sequence:
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ……..
Then the total number of children is at least 12 and, from (iv), at most 24. Also, if the total number of children is even, A must have an odd number of children, and if the total number of children is odd, then A must have an even number of children.
The total number of children cannot be 13 because, no three numbers, one from each sequence can give us a total of 13. The total cannot be 12, 14, 15, 16 or 17 because then the number of children each had would be known, contradicting (iv). The total cannot be 18, 20, 21, 22, 23 or 24 because then no number of children could be known for anybody, again contradicting (iv). So, the total number of children has to be equal to 19.
When the total is 19, A must have an even number of children and from the sequences, this number must not be greater than 19 – (4+5), which is equal to 10. So, A must have 6 children. Then B and C together must have 13 children. Then B must have either 4 or 8 children. Now, if B has 4 children, then it is clear that C has 9 children and if B has 8 children, then C has 5 children.
So, now it is clear that the speaker is ‘A’.
