Henry Ernest Dudeney/Puzzles and Curious Problems/127 - Find the Squares/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $127$

Find the Squares
What is the number which, when added separately to $100$ and $164$, make them both perfect square numbers?


Solution

$125$


Proof

In this context Dudeney, who is writing primarily for a non-mathematical audience, is looking for a strictly positive integer.

In puzzles such as this, where his audience consists of readers of popular journals, the concept of negative numbers will not be considered unless specifically prompted and primed.


We are looking for two square numbers whose difference is $64$, that is, $31 + 33$.

We have that:

$31 = 2 \times 15 + 1$
$33 = 2 \times 16 + 1$

By the Odd Number Theorem, those two square numbers are $15^2$ and $17^2$.

Thus we have:

$15^2 = 225$
$17^2 = 289$

Hence the number that needs to be added to $100$ and $164$, as required, is $125$.

$\blacksquare$


Sources

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