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\begin{document}

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\section*{subdivided circles solution}

\hypertarget{solution_to_the_subdivided_circles_puzzle}{}\section*{{Solution to the [[Subdivided Circles]] Puzzle}}\label{solution_to_the_subdivided_circles_puzzle}

[[SubdividedCircles.png:pic]]

\begin{quote}%
What fraction of each circle is shaded? The $12$ dots are equally spaced, and the only point used inside the circle is the centre.


\end{quote}
\hypertarget{solution_by_dissection}{}\subsection*{{Solution by [[Dissection]]}}\label{solution_by_dissection}

Each of the diagrams can be dissected to show that the total shaded area is half of that of the circle.

[[SubdividedCirclesFurtherDivided.png:pic]]

In the first three circles (the yellow, pink, and light blue), the regions can be rearranged as labelled to show that half of each circle is shaded.

For the green circle, the central dart needs to be split into three regions. These combine with the outer segments as shown to show that half the circle is shaded.

The dark blue circle needs two stages. The dissection applies to each of the three strips in the same way so only one is shown. In the first step, the region labelled $1$ is reflected to make the region on the left in the above diagram. This is then further dissected as on the right in the same diagram: the region labelled $4$ is moved from the edge to the centre. The result is a sector of the circle that is one sixth of the whole circle, so three of these comprises half the circle.

The orange circle \ldots{} is a work in progress!



\end{document}