John is older than Mary. He notices that if he switches the two digits of his age (an integer), he gets Mary's age. Moreover, the difference between the squares of their ages is the square of an integer. How old are Mary and John?
Let John's age be $10 a+b$ where $0 \leq a, b \leq 9$. Then Mary's age is $10 b+a$, and hence $a>b$. Now

$$
(10 a+b)^{2}-(10 b+a)^{2}=9 \cdot 11(a+b)(a-b) .
$$

Since this is the square of an integer, $a+b$ or $a-b$ must be divisible by 11 . The only possibility is clearly $a+b=11$. Hence $a-b$ must be a square. A case study yields the only possibility $a=6, b=5$. Thus John is 65 and Mary 56 years old.