Number of regions formed by zigzags
Source: India tst 2003, p6
March 14, 2012
algebrapolynomialcombinatorics unsolvedcombinatorics
Problem Statement
A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find , the maximum number of regions into which zig-zags can divide the plane. For example, (see the diagram). Of these regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in of degree not exceeding .
[asy]
draw((30,0)--(-70,0), Arrow);
draw((30,0)--(-20,-40));
draw((-20,-40)--(80,-40), Arrow);
draw((0,-60)--(-40,20), dashed, Arrow);
draw((0,-60)--(0,15), dashed);
draw((0,15)--(40,-65),dashed, Arrow);
[/asy]