# Counter-intuitive properties of infinite sets

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Counter-intuitive properties of infinite sets for a lay audience – SURPRISES IN SUM OF INFINITE SETS

you will try to describe some counter-intuitive properties of infinite sets for a lay audience.^{[if !supportFootnotes][1][endif]} Think of as an article that might appear in a popular magazine devoted to science or as an article in a Sunday paper. You may assume your audience has a background in precalculus mathematics but no calculus.

The Blocks Unlimited store sells various sets of toy blocks. One set, called the Deluxe Set, consists of infinitely many cubes, the first of which is 1 ft. by 1ft. by 1ft., the second cube has edge length 1/2 ft, the third has edge length 1/3 ft, and the n^{th }cube has edge length 1/n ft. A second set of blocks, called the Starter Set, is a subset of the Deluxe Set. It consists of infinitely many cubes, the first of which has edge length 1 ft, the second has edge length 1/2 ft, the third has edge length 1/4 ft, and the n^{th} cube in the Starter Set has edge length 1/2^{n-1 }ft.

Give a convincing argument that if all the blocks in the Deluxe Set were stacked one on top of the other, then the stack would extend beyond Alpha Centauri but that it is possible to pack the Deluxe Set into a box that would be small enough to easily fit inside the trunk of a sports car. Since the Starter Set is a subset of the Deluxe Set, it could be packed in the same box used for the Deluxe Set. Argue that if the cubes in the Starter Set were stacked one on top of another, then the stack would be not very high at all and the exact height of this stack can be determined.

The examples given in the previous paragraph are to be discussed in detail in 1. You should include some additional examples of your own creation or a discussion of one or more of the following topics:

- a) The cubes in the sets are sold unpainted. The Blocks Unlimited store also sells a special paint that can be used to paint these cubes. The paint is special because it has zero thickness. This paint is sold by the square foot. Compute how many square feet of paint one would need to buy in order to paint all the faces of all the cubes in the Starter Set.

- b) Estimate how many square feet of paint one would need to buy in order to paint all the faces of all the cubes in the Deluxe Set.

- c) If instead of painting the entire cube, suppose only a thin stripe is painted on one edge of each cube of the Deluxe Set. Now suppose the cubes are stacked one on top of another so the stripes along the edges lineup. As you argue, this stripe would be arbitrarily long. Estimate how many square feet of special paint would be needed to paint this stripe. Would an infinite number of square feet of paint be needed?

- d) Can you explain the apparent paradoxes?

You may assume your reader knows basic algebra and will remember, when reminded, formulas for geometric series. However, the reader does not know about definite integrals or about divergent series. Thus, the challenge in writing is to justify the claims about these sets of blocks using only the mathematics the reader already knows.

This assignment is a minor variant on earlier assignments of Baldwin, Berman and Radford and is based eventually on an exercise in Writing in the teaching and learning of mathematics, J. Meier and T. Rishel, MAA 1998.