Sciolism Rocks

In my last blog post, I showed how the second Google Treasure Hunt problem led to Pascal’s Triangle, in an effort to show how some problem solving strategies and a little intuition can lead to an easy problem solution.

I left you with a problem to work on yourself.

This post reveals the answer to this problem (it’s not what you think) and “magically” connects it to the original Google treasure hunt problem.

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From the original picture, you could fill in the first three elements of this table:

Number
of Points		Regions
   2			   2
   3			   4
   4			   8
   5			   _
   6			   _
   7			   _ <-This is the number I wanted

I went ahead and drew the circles and points for the seven circles, and counted the regions that appeared.

Most of the responses I received drew and counted the circles for the five-point problem. Seeing the pattern with just that information looks like this:

Number
of Points		Regions
   2			   2
   3			   4
   4			   8
   5			   16
   6			   _
   7			   _

…which made the problem look easy, and even intuitive (!). Adding another slice through the circle must double the number of regions, so, there must be 32 regions for 6 points and 64 regions for 7 points. However, if you look back at my picture, you’ll see that initial solution is incorrect: 32 regions and 64 regions are impossible. The picture shows that you can’t get more than 31 regions out of 6 slices, nor more than 57 out of 7.

Number
of Points		Regions
   2			   2
   3			   4
   4			   8
   5			   16
   6			   31
   7			   57

So we’re now a bit lost. Is there a pattern in these numbers at all ? I’ll use JMP to graph the Regions vs. Number of Points:

There appears to be a pattern. I can use JMP to fit several polynomials, to see which one looks best. Following are pictures of a quadratic, cubic, and quartic polynomial fits.

The quartic polynomial is exactly the right fit. And JMP tells me that the relationship between Regions (y) and Number of Points (x) is

Using this formula, you can get a prediction for the number of regions using 8 points.

Which works out to be 99 regions. If you want to count them to verify there are 99, go ahead:

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The point of this (admittedly, long) exercise in circles, lines, and points, is that your intuition may serve you well (as in the original Amanda-based problem), or it might lead you astray (as in the simple answer to the circles problem). Observing a pattern may lead you to good results, but you often have to have more information than you think to observe the correct pattern.

There’s one more punch line I owe you: How does the Amanda problem match up with the circles problem?

remember the key to solving the Amanda problem was found in Pascal’s triangle:

The solution to the circles problem was this:

Number
of Points		Regions
   2			   2
   3			   4
   4			   8
   5			   16
   6			   31
   7			   57
   8             99

Do you see the pattern for the number of regions in Pascal’s triangle? Here, let me help. If you add up the sums of the rows, you get the original (incorrect) solution to the circles problem.

Do the same thing : add up the sums of the rows, but only to the left of the orange line. Now you’ve got the real answer to the circles problem.

That’s right, both problems (and a lot more) are solved using the same pattern of numbers.

What’s not to like about math?