Confusing Ages in a FamilyNumber Theory Level 3
In a family of four, the mother is three times as old as the daughter was when the father was the same age as the mother is now. When the daughter reaches half the age the mother is now, the son will be half as old as the father was when the mother was twice the age the daughter is now. When the father reaches twice the age the mother was when the daughter was the same age as the son is now, the daughter will be four times as old as the son is now. Given that one of their ages is a perfect square, what is the sum of the four ages?