Maximum Circle Intersections

Probability Level 1

What is the maximum possible number of points of intersection of 6 distinct circles?

The answer is 30.

2 solutions

Sahil Bansal
Feb 12, 2016

We can select any 2 circles from the 6 circles and their maximum number of points of intersection will be 2. Hence, the total number of points of intersection can be expressed as follows:

$2 \times \begin{pmatrix} 6 \\ 2 \end{pmatrix} = 2 \times \cfrac { 6 \times5 }{ 2 }=\boxed{30}$

Though this established the upper bound, this may not be the maximum unless we can show its existence. for example the maximum number of intersection of N chords of a circle is non trivial

Cheng Wei Chang - 5 years, 3 months ago

Hi @Sahil Bansal, if you type plain text in LaTeX it doesn't look very nice - instead, you can type

\text{insert text here}

which produces $\text{insert text here}$ , if you want to type something in LaTeX. Cheers!

Michael Fuller - 5 years, 4 months ago

$\text{Thank you. I will follow this method from next time.}$

Sahil Bansal - 5 years, 4 months ago

Oli Hohman
Dec 19, 2016

I noticed a pattern when analyzing lower cases and found that the maximal number of intersections of distinct circles is actually a quadratic sequence

2 circles --> 2 intersections 3 circles--> 6 intersesctions 4 circles --> 12 intersections 5 circles --> 20 intersections 6 circles --> 30 intersections

From this, I assume that n circles ---> n^2-n maximal intersections

Is the above true in general? Can anyone use combinatorics to prove it, if so?

Maximum Circle Intersections

The answer is 30.

2 solutions

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