1, 2, 4, 8, what comes next?

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Suppose you take a circle, put some dots along the outside, and then connect them, as in the picture to the left (which has 5 dots on the outside). If only two lines cross at any point, how many regions will the circle be divided into?

Let’s find out! If you place 1 dot along the outside you can’t connect it to anything so you get one region (the entire circle). If you place 2 dots along the outside and connect them you divide the circle into 2 regions. By placing 3 dots along the outside you divide the circle into 4 regions.

Looking at the bottom row, if you place 4 dots on the circle and connect them you get 8 regions. Hey, notice that neat 1, 2, 4, 8 pattern of the title? What about five dots?

At this point it’s getting a little hard to keep track of all the regions, so I colored them in the pictures below.

Looking at the middle of the bottom row, putting 5 dots along the edge divides the circle into 16 pieces, as might be expected.

What if there are 6 dots along the outside? And that’s where everything goes terribly wrong. It only divides the circle into 31 pieces, not 32. Putting 7 dots along the outside is even worse: you get 57 pieces.

It turns out that even though the numbers 1, 2, 4, 8, 16, 31 start off as powers of 2, they’re really following the formula \frac{1}{24} \cdot (n^4 - 6{n^3} +23{n^2} - 18n  + 24), where n is the number of dots on the outside. (Where the heck does that come from, you might wonder? It actually comes from the simpler-looking {n \choose 4}+{n \choose 2}+1 (see Wolfram’s Mathworld)).

So much for our nice pattern.

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