22 Aug 2013

Partitioning a Circle: Upper Secondary Mathematics Competition Question

This is the last of this week’s excursions into pizzas and cakes. In essence, these questions have been about the partitioning of two- and three-dimensional figures, so let’s abandon the culinary analogies for this one.

Take a circle and distribute n distinct points around its circumference. Join each point to every other point with a chord. The circle has thus been partitioned into a number of non-overlapping regions. Let R(n) be the maximum number of regions that can be created.

Given that R(n) is a quartic polynomial, or otherwise, find R(n) and hence calculate R(10).

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