Problem 3. (20 points) The plane is divided into regions by straight lines. Show by mathematical induction that it is always possible to color the regions with 2 colors so that adjacent regions are never the same color.

Figure 1: Lines drawn in plane. The regions in the plane created by these lines can be colored with two colors in such a way that any two regions sharing a common border have dierent colors. Hint 1: Use induction on the number of lines.

Suggestion: Approach this problem by drawing a few lines and coloring the regions.

Hint 2: Mathematical induction is a proof method closely related to algorithmic proofs. Suppose the plane is divided into regions by k lines and is colored with two colors as the problem asks. Now suppose you draw a k + 1-st line in the plane. This new line will create a few new regions. Can you think of a procedure that assigns colors to the regions so that adjacent regions are still colored with two colors?

Problem 3. (20 points) The plane is divided into regions by straight lines. Show by mathematical induction that it is always possible to color the regions with 2 colors so that adjacent regions are never the same color Figure 1: Lines drawn in plane. The regions in the plane created by these lines can be colored with two colors in such a way that any two regions sharing a common border have different colors. Hint 1: Use induction on the number of lines. Suggestion: Approach this problem by drawing a few lines and coloring the regions. plane is divided into regions by k lines and is colored with two colors as the problem asks. Now suppose you draw a k +1-st line in the plane. This new line will create a few new regions. Can you think of a procedure that assigns colors to the regions so that adjacent regions are still colored with two colors? Problem 3. (20 points) The plane is divided into regions by straight lines. Show by mathematical induction that it is always possible to color the regions with 2 colors so that adjacent regions are never the same color Figure 1: Lines drawn in plane. The regions in the plane created by these lines can be colored with two colors in such a way that any two regions sharing a common border have different colors. Hint 1: Use induction on the number of lines. Suggestion: Approach this problem by drawing a few lines and coloring the regions. plane is divided into regions by k lines and is colored with two colors as the problem asks. Now suppose you draw a k +1-st line in the plane. This new line will create a few new regions. Can you think of a procedure that assigns colors to the regions so that adjacent regions are still colored with two colors