How to approach problems

Place $n$ points on a circle, and connect each pair of points by a line segment. What is the maximum number of regions we obtain inside the circle?

Start by stating and thinking about some examples.

• 1 point $\Rightarrow$ 1 region
• 2 points $\Rightarrow$ 2 regions
• 3 points $\Rightarrow$ 4 regions
• 4 points $\Rightarrow$ 8 regions

With this pattern we notice that $R$ being the number of regions given by $n$ points on a circle, we get the relation: $R(n) = 2^{n-1}$

Once you see a pattern, try to prove it

If we go for $n=6$ we see that $R(n) = 31$. Therefore, our hypothesis is incorrect. There actually isn't a function to describe the above problem.

Remember to try using small numbers as examples to confirm your intuition.