Need a hint?

I want to color the design below so that each region is a given a single color and so that no two neighboring regions, that is, those sharing a positive length of boundary, are given the same color. (Two regions that meet at a point, however, may be given the same color.)

The following picture uses six different colors, but I am sure I could accomplish this feat with fewer.

What is the least number of different colors one could use to successfully color this design?

Scroll down for the solution to the original problem.

Click here for the solution to the Challenge Problem

The answer is zero. Imagine that runners are running along the track in the direction indicated by the arrows. If you were to cross the track, walking from one region to another, then runners will either be running from your left to your right as you cross (call this a "positive" perspective) or from your right to your left (the "negative" perspective). Now imagine standing in the region labeled "2". To leave this region, you must cross the track twice, both in the positive sense – hence the label "2". Escaping from any region labeled "1" requires crossing the track once with the positive sense and escaping from the region labeled “-1” requires crossing the track once with the negative sense. To answer the puzzle … notice that leaving the region in question requires crossing the track twice, once in the positive sense and once in the negative sense. The total count is: 1 + (-1) = 0. Consequently, this region should be labeled zero!

SOMETHING CURIOUS: Draw any wiggly path from any point inside the region labeled "2" to any point outside the track. Draw one that crosses the track multiple times and then count +1 each time you cross the track with the runners approaching you from the left, the positive sense, and -1 for each time the runners approach you from the right, the negative sense. The sum of all these numbers is always two! (Really do try this. Experiment with paths that spiral and circle many times, cross back and forth inefficiently, and even retrace themselves.)

To see why the label of any region is independent of the path you follow to escape it, click here.

If you attempted the Challenge Puzzle, click here for the solution and explanation.