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. Click for colored drawings showing this, and also demonstrating that the number of regions does NOT continue to double!