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## 'Euler's Squares' printed from http://nrich.maths.org/

Euler found four whole numbers such that the sum of any two of the
numbers is a perfect square. Three of the numbers that he found are
\begin{equation*} a= 18530, \quad b=65570, \quad c=45986.
\end{equation*}

Find the fourth number, $x$. You could do this by trial and error
(sometimes called trial and improvement), and a spreadsheet would
be a good tool for such work. However, Euler would not have used
any electronic calculating aids to find his 'fearsome foursome' and
he would have found ways of reducing the search to a small number
of cases and this is what you should try to do. You could do this
by writing down \begin{equation*} a+x = P^2 \end{equation*}
\begin{equation*} b+x = Q^2 \end{equation*} \begin{equation*} c+x =
R^2, \end{equation*}

and then focussing on $ Q^{2}-R^2=b-c $ which is known. Moreover
you know that $ Q > \sqrt{b} $ and $ R> \sqrt{c} $. Use this
to show that $ Q-R \leq 41 $. Use a spreadsheet to calculate values
of $ Q+R $, $ Q$ and $ x $ for values of $ Q-R $ from $ 1 $ to $ 41
$, and hence to find the value of $ x $ for which $ a+x $ is a
perfect square.

There may be better ways to do this, and if you find one, do let us
know!