KING = Big Portrait of himself and slightly small portrait of the Queen.
QUEEN = Big Portrait of herself and extremely small portrait of the King.
TOTAL = 1865cm
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Your Answer :
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$$x^2+y^2=1865$$
We can solve this with just a little sensible trial and error.
$${\sqrt{{\mathtt{1\,865}}}} = {\mathtt{43.185\: \!645\: \!763\: \!378\: \!368\: \!2}}$$
So for the 2 that are a long way apart one is are going to be just a little smaller than this number and the other will be little.
Try 43
$${\sqrt{{\mathtt{1\,865}}{\mathtt{\,-\,}}{{\mathtt{43}}}^{{\mathtt{2}}}}} = {\mathtt{4}}$$ ok so the first 2 queens ones are 4cm and 43cm
Now lets think about the 2 that are close together.
think about
$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}} = {\mathtt{1\,865}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{1\,865}}}}}{{\sqrt{{\mathtt{2}}}}}}\\
{\mathtt{x}} = {\frac{{\sqrt{{\mathtt{1\,865}}}}}{{\sqrt{{\mathtt{2}}}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{30.536\: \!862\: \!969\: \!204\: \!940\: \!9}}\\
{\mathtt{x}} = {\mathtt{30.536\: \!862\: \!969\: \!204\: \!940\: \!9}}\\
\end{array} \right\}$$
Bingo, the Kings ones are 29cm and 32 cm
The King's pictures were 32cm x 32cm and 29cm x 29cm
and
The queen's were 43cm x 43cm and 4 cm x 4cm
