Square Cutting

The average number of regions into which $N$ lines divide a Square is

$${\textstyle{1\over 16}}N(N-1)\pi+N+1$$

(Santaló 1976).

See also Circle Cutting

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/geom/geom.html

Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.