Hello all, I've been trying to work out this problem I came across yesterday when a professor mentioned it in a math education course. It states, "Take a circle and put two dots on the circle then connect them with a cord. How many sections of area does the cord split the circle into?" Of course the answer is 2. Now do it with 3 dots. The answer is 4. Now do it with 4 dots. The answer is 8. Now do it with 5 dots. The answer is 16. How many if there are 6 dots? One might think, naively as I did, that the formula for the number of sections is 2n-1, so 6 dots should have 32 sections. But it actually has 31. That was the point of the lesson. We never got the formula so I've been trying to figure one out. I decided to look at the number of dots and how many cords they form, and I was able to find a recursive definition and conjecture a formula. I proved it using induction. I will write the information out like this: # of dots / #of cords; 1 / 0 2 / 1 3 / 3 4 / 6 5 / 10 6 / 15 7 / 21 8 / 28 The recursive series can be understood as s(n)=s(n-1) + (n-1), and the explicit formula is: s(n)=(n-1)+(n-2)+...+ 3+2+1. Now I have a nice relationship between dots and cords. Next I will show you the table of dots to cords to sections, which I worked out manually up to n=7. 1 / 0 / 1 2 / 1 / 2 3 / 3 / 4 4 / 6/ 8 5 / 10 / 16 6 / 15 / 31 7 / 21 / 58 8 / 28 / ??? So this is where I am stuck. I am trying to find a relationship with the number of sections and the number or cords or dots. I do not see any obvious recursive relationship, and certainly do not see any obvious formula. Perhaps a different set of eyes/synapses on the problem would shed more light. Is there a more systematic approach I can use, rather than jus trying to "see" it?