By an n-fold line subdivision of the plane P, we mean any collection of n-distinct (infinite) lines in P, together with the open regions in P that they determine. (We don't count the lines as part of the regions.) Let us say that two such regions are adjacent if their boundaries have a positive-length or infinite line segment in common. Prove the following theorem by induction on n:
Given any n-fold line subdivion of P, each of the regions in the subdivision may be colored either red or blue in such a way that no two adjacent regions have the same color.
(Hints: You may also use (without proof) the following facts: (1) Every infinite lines in P divides P into two regions of which it is the common boundary. (2) Given an open region R in a k-fold line subdivision of P, if one introduces a new line L distinct from the others, then either L does not meet R or L subdivides R into two adjacent regions with common boundary segment contained in L.)
Hello and thank you for posting your question to Brainmass!
The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format ...
Fold lines are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.