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Sunday Afternoon Maths XLI

 Posted on 2015-04-05 

Squared circle

Each side of a square has a circle drawn on it as diameter. The square is also inscribed in a fifth circle as shown.
Find the ratio of the total area of the shaded crescents to the area of the square.

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Polygraph

Draw a regular polygon. Connect all its vertices to every other vertex. For example, if you picked a pentagon or a hexagon, the result would look as follows:
Colour the regions of your shape so that no two regions which share an edge are the same colour. (Regions which only meet at one point can be the same colour.)
What is the least number of colours which this can be done with?

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If you enjoyed these puzzles, check out Advent calendar 2019,
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