Take a circle:

(1) One line divides the circle into 2 regions

(2) 2 lines divides the circle into 4 regions as long as the lines intersect in the circle.

(3) 3 lines can divide the circle into 7 regions. The third line must intersect both of the pervious lines...

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Take a circle:

(1) One line divides the circle into 2 regions

(2) 2 lines divides the circle into 4 regions as long as the lines intersect in the circle.

(3) 3 lines can divide the circle into 7 regions. The third line must intersect both of the pervious lines in distinct points -- i.e. not through the point of intersection.

** If you draw this you will notice a region from step 2 that is untouched. This always happens. **

From now on the rule is that any additional line must intersect all previous lines, and cannot go through an old intersection point.

(4) 4 lines can divide the circle into 11 regions.

(5) 5 lines gives 16 regions.

You should notice a pattern. For line number "n" there are "n" additional regions created. For n=2 there were 2 regions, now 4. For n=5 there were 11 regions now 11 etc...

This appears to be quadratic. You can use the method of finite differences to add evidence if you wish. Using regression or solving a system of equations using any three points (such as (1,2),(2,4) and (3,7)) we find the quadratic to be `f(n)=1/2(n^2+n+2)`

In particular, for n=11 we get `f(11)=1/2(11^2+11+2)`

`f(11)=1/2(121+11+2)=1/2(134)=67`