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

by : $127$

 * Find the Squares

Solution

 * $125$

Proof
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 $125$.