Henry Ernest Dudeney/Modern Puzzles/21 - Their Ages/Solution

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Modern Puzzles by Henry Ernest Dudeney: $21$

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:

\(\ds T^2 + M\) \(=\) \(\ds 62\)
\(\ds M^2 + T\) \(=\) \(\ds 176\)
\(\ds \leadsto \ \ \) \(\ds M\) \(=\) \(\ds 62 - T^2\)
\(\ds \leadsto \ \ \) \(\ds \paren {62 - T^2}^2 + T\) \(=\) \(\ds 176\)
\(\ds \leadsto \ \ \) \(\ds T^4 - 124 T^2 + T + 62^2 - 176\) \(=\) \(\ds 0\)

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.

$\blacksquare$


Sources

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