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Circle and Points Problem and Coronavirus Update

Circle and Points Problem

Last post I posed this problem:

Place n points on a circle and draw all the line segments that connect any two of the points. The points are placed such that no three of the segments intersect at a single interior point. Into how many regions do the segments divide the circles interior?

If you start to tabulate the answers for small values of n, you find:

n12345678
Regions1248????

It looks like the solution to this problem is that the number of regions follows the pattern of the powers of 2.

If you draw the figure for 5 points, you will find as expected the number of regions is 16. But when you draw the figure for 6 points:

— no matter where you place 6 points around the circle, the number of regions is 31, not 32!

This problem demonstrates that finding a pattern is not as simple as examining the first 3 or 4 or 5 cases. Ultimately, it will be necessary to prove that any pattern you find is in fact the solution to the problem.

Coronavirus Update

I’m continuing to track the number of cases in Santa Clara county. The latest graph looks like:

As you can see if the rate of change of the daily case count continues its trend, the number of cases will start to level off in two weeks. That would be great news!

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