This pattern breaks, but for a good reason | Moser's circle problem

This is a very famous cautionary tale in math, known as Moser's circle problem.

Some of you may have seen this before, but what I

want to do here is really explain what's going on.

The way this starts is we take a circle and put two points on that

circle and connect them with a line, that is a chord of the circle,

and note that it divides the circle into two different regions.

If I add a third point and then connect that to the previous two points with two

more chords, then these lines all divide the circle into four separate regions.

Then if you add a fourth point and connect that to the previous three,

and you play the same game, you count up how many regions has this cut the circle into,

you end up with eight.

Add a fifth point to the circle, connect it to the previous four,

count up the total number of regions, and if you're careful with your counting,

you'll get a total of sixteen.

Naturally, you can guess what might come next, but would you bet your life on it?

Add a sixth point, connect it to all the previous ones,

and if you carefully count up all the different regions,

you end up not with the power of two you might have expected, but just one shy of it.

Some of you might be raising your hand saying,

doesn't it depend on where we put the points?

For example, watch how this middle region disappears if I

place everything nice and symmetrically around the circle.

So yes, it does depend, but we're going to consider the cases

where you never have any three lines intersecting with each other.

This would be the generic case if you just choose n random points,

almost certainly you'll never have three lines coincide,

but setting aside the technical nuances, the problem is such a tease,

it looks so convincingly like powers of two until it just barely breaks.

And I have such a strange soft spot for this particular question.

When I was younger I wrote a poem about it and also a song.

And on the one hand it's kind of silly, because this is just one example of what

the mathematician Richard Guy called the strong law of small numbers,

summed up in the phrase, there aren't enough small numbers to meet the many demands

made of them.

But I think what I really like about this problem is that if you sit down to try

to work out what is the real pattern, what's actually going on here, one,

it's just a really good exercise in problem solving,

so it makes for a nice lesson right here, but also it's not just a coincidence

that it starts off being powers of two.

There's a very good reason this happens.

And it's also not a coincidence that you seemingly randomly hit

another power of two a little bit later on the tenth iteration.

So we've got this pattern, and what you want to find is what function describes it.

If you put n points on the boundary of a circle,

and you connect them with all the possible chords,

and you count how many regions the circle has been cut into,

if the answer isn't a power of two, what is it?

What function of n should we plug in?

As always with math, problem solving rule number one if you're stuck is

to try solving easier questions somehow related to the problem at hand.

It helps you get a foothold, and sometimes those

answers are helpful in the final question.