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\begin{document}

%-------------------------------------------------------------------

\section*{square overlapping a square and a rectangle solution}

\hypertarget{solution_to_the_square_overlapping_a_square_and_a_rectangle_puzzle}{}\section*{{Solution to the Square Overlapping a Square and a Rectangle Puzzle}}\label{solution_to_the_square_overlapping_a_square_and_a_rectangle_puzzle}

[[SquareOverlappingaSquareandaRectangle.png:pic]]

\begin{quote}%
The area of the largest square is $25$. What’s the combined area of the smaller square and rectangle?


\end{quote}
\hypertarget{solution_by_invariance_principle}{}\subsection*{{Solution by [[Invariance Principle]]}}\label{solution_by_invariance_principle}

[[SquareOverlappingaSquareandaRectangleCaseOne.png:pic]]

The angle of the larger square can be varied. When it is aligned horizontally, the outer square and the darker blue square coincide (meaning that the lighter blue rectangle effectively does not exist) showing that they have the same area.

[[SquareOverlappingaSquareandaRectangleCaseTwo.png:pic]]

Alternatively, if the larger square is tilted at $45^\circ$ to the horizontal then the lighter blue rectangle is also a square and the two shaded squares have the same combined area as the outer square.



\end{document}