5 - Colored Marbles

Summary

A useful strategy in dealing with smoothing transformations is to pay special attention to the maximum and the minimum of the set of numbers being transformed. For example, in Chapter 2, the arithmetic–geometric mean inequality was proved by means of a carefully designed transformation which replaces the maximum and the minimum of a set of positive numbers by two numbers lying between them, while the rest of the numbers remain unchanged. The following example, based on a problem from the Invitational Mathematical Competition of three provinces in Northeastern China, 1987, further illustrates the usefulness of this strategy.

Given a box containing an unlimited supply of red, yellow, and blue marbles, select at random a set of 1,987 marbles from the box. Define a trading operation in which any two differently colored marbles from the selected set can be traded for two marbles of the third color from the box.

• Show that no matter how the colors are initially distributed, it is always possible to perform a finite number of trades which will result in 1,987 marbles of the same color.

• Based on the initial distribution of colors, predict what the final color must be.

To prove the first statement, let b, r, and y be the number of blue, red, and yellow marbles in the set. Note that b + r + y = 1,987. Note that the case for which b = r (or b = y or r = y) has the following simple solution.