Circle Dots

A circle and many dots?

Consider a circle with n dots on its circumference; line segments join every dot to every other dot; no point, except the dots themselves; lies on more than two line segments. Find a formula for the number of regions in the circle (i.e. if there are 6 dots on the circumference of the circle, how many regions will be formed when we connect every six dot with one another?)

This is a really interesting question and one I always gave my students years ago. This is because if you find the number of regions for 1, 2 , 3 ,4 , 5 dots the answers are 1, 2, 4, 8, 16 and so everyone is convinced that the next result for 6 dots will be 32 but it isn’t it’s 31. Frequently students thought that they must have miscounted and looked for the missing region. However 7 dots giving 57 regions made them realise that the formula is not 2^(n – 1). In fact the formula is nC4 + nC2 + 1. Here C is standing for the number of combinations. For n > 3 it can be written as n!/(24(n – 4)!) + n!/(2(n – 2)!) + 1. Proving it is not that easy but possible. The moral of the story is not to think that you’ve found a formula too quickly.

P.S. What are the last two on about?

Circle Circle Dot Dot (funny video)

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