The Circle-Dots problem provides a cautionary tale to those who rely solely on patterns they observe (and, more importantly, the assumption that these perceived patterns must continue).

The idea of the problem is fairly simple.

- Place a certain number of dots on the circumference of a circle.
- Connect each dot with each of the remaining dots with a line segment.
- Count the number of regions into which the segments partition the circle’s interior.

As seen below, 2 dots partition the circle into 2 regions, 3 dots partition the circle into 4 regions, and 4 dots partition the circle into 8 regions. It might seem that an obvious conjecture is that 5 dots should partition the circle into 16 regions. As shown below, that is indeed the case.

It seems that adding a dot always doubles the number of regions, right? In fact, you might even guess that n dots will partition the circle into 2^{n-1} regions. What could be more obvious? Well, this example is a good one to remind us that “obvious” patterns don’t have to be true! To see why, check to see how many regions 6 dots partition the circle into. Not only is the answer NOT 32, there’s not even just one possible answer!! You’ll always either get 30 or 31 regions, depending on how the dots are arranged around the circle:

SO, the next time you think to yourself, “This pattern HAS TO continue!”, remember the Circle-Dots problem. Sometimes we see the “wrong” pattern and sometimes there isn’t a pattern AT ALL!