Here's a puzzle full of clashing colors. Picture a geometric figure that's colored partly red and partly blue. If the figure contains at least one red point and one blue point that are exactly 10 centimeters apart, mathematicians would say--in their wonderfully abstruse jargon--that the figure satisfies the 10-centimeter red-blue bicoloration condition. The idea behind this month's puzzle is to take a simple shape, such as a line segment or a circle, and determine whether every conceivable pattern of red and blue on that figure satisfies the 10-centimeter condition.

As a warm-up, consider line segments. Can you paint a red-and-blue pattern on a line segment that does not satisfy the 10-centimeter condition? Assume that the line segment is one meter long, with the zero-centimeter mark at the left end and the 100-centimeter mark at the right (illustration A). Color the following intervals red: 0-1, 9-11, 19-21, 29-31, 39-41, 49-51, 59-61, 69-71, 79-81, 89-91 and 99-100. Now color all the other intervals blue. Notice that every point on this segment will have the same color as the points that are 10 centimeters to its left or right. So this pattern does not satisfy the 10-centimeter condition.

This article was originally published with the title Seeing Red, Feeling Blue.

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