My All-Time Favorite Math / Geometry Puzzle

This past weekend I took a walk from my home to the local supermarket, just to get some exercise. When I got there, I headed over to the magazine rack and saw a collection of puzzles.

As I browsed the magazine, I saw a puzzle that instantly reminded me of my all-time favorite puzzle. If you place n dots on the circumference of a circle, and connect all n dots with lines to every other dot, how many line intersections do you get?

For example, if n = 5, there are 5 intersections formed. And if n = 6, there are 15 intersections. Is there a general rule? If so, what is it?

This puzzle was first posed to me by one of my college professors, Mike Butler, when I was an undergraduate at the University of California at Irvine. The class met on Tuesdays and Thursdays, and Butler gave us the question on a Thursday. The class had really smart people in it, and we were all competitive so I really wanted to be the only student who solved the problem.

I worked on the problem all weekend, and all day Monday, but just couldn’t come up with the solution. On Tuesday morning, I was riding the bus from my house on Balboa Island to class (with my roommate Ed Koolish who was also in the class) and was feeling quite defeated when the answer suddenly came to me in a flash of insight. I can still remember exactly how I was sitting on the bus and the exact position of my right arm on the armrest.

The moral of the story is that our subconscious is always working.

I’ve put the answer to the puzzle below.

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2 Responses to My All-Time Favorite Math / Geometry Puzzle

  1. greg7mdp says:

    Why do the point have to be on a circle? That does not change the number of line intersections.

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