Another Daughters' Ages Quiz Puzzle Answer

We meet some odd characters in The Jolly Quizmaster. Only the other day there was a lady talking about her three daughters. This lady must have had a liking for mathematics, as she told the customers the following about her daughters' ages:

The sum of the products of Ann's age and Beth's age, of Beth's age and Carol's age, and of Carol's age and Ann's age (all ages in whole years) is 37 more than the corresponding total last year.

And the sum of the products this year is equal to the square of Ann's age next year.

Naturally we took out a paper and pencil and did the sums. So how old are the three daughters?

Ann is ten and the other two girls are seven and three, although we have no way of knowing which way round.


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