Henry Ernest Dudeney/Modern Puzzles/21 - Their Ages

by : $21$

 * Their Ages
 * If you add the square of Tom's age to the age of Mary,
 * the sum is $62$;
 * but if you add the square of Mary's age to the age of Tom,
 * the result is $176$.


 * Can you say what are the ages of Tom and Mary?

Solution
Tom is $7$ and Mary is $13$.

Proof
Let $T$ be the age of Tom and $M$ be the age of Mary.

Expressing one variable in terms of another and eliminating one of them results in a quartic:

Good luck with that.

So, let us assume that both $T$ and $M$ are natural numbers.

It is apparent that $T < 8$ and $M < 14$, otherwise the totals will be such that either $T$ or $M$ will be negative.

So:
 * $176 - M^2 < 8$

which leads us to:
 * $M^2 = 169$

giving us:
 * $M = 13$

and:
 * $T = 176 - 169 = 7$

We check the other equation:
 * $7^2 + 13 = 49 + 13 = 62$

and all is well.