December 21, 2000

**Old Challenge **(Timur Dogan). Which branches of mathematics or types of problems are the most counterintuitive?

**Answer** (Eric Brahinsky). The mathematics of infinity, especially that there are different sizes of infinity. The infinity of all positive integers is the same as the infinity of even positive integers, because you can pair them up, matching n to 2n. The infinity of all real numbers is bigger: you can prove that there is no way to match them up with the positive integers. Jonathan Falk notes, however, that the infinity of rational numbers or fractions, however, turns out to be the same as the infinity of positive integers.

An infinite series of numbers can have a finite sum, such as

1/2 + 1/4 + 1/8 + 1/16 + . . . = 1.

As usual with addition, for this series the order in which you add the terms does not matter. For other series, however, rearranging the order of the series can change the answer.

Falk has a few other favorites:

1. Goedel's proof that in any reasonable logical system of mathematics, there will be true facts which cannot be proved from the axioms.

2. Arrow's theorem that the only reasonable voting system is a dictatorship.

3. The St. Petersburg Paradox that the following betting game is worth infinitely much: If your first n tosses of a coin are heads you win 2^{n} dollars.

Carl Eichenlaub is amazed that in a room of only 23 people, the odds are that some two of them will have the same birthday.

Joe Shipman likes the Banach-Tarski paradox, which says for example that you can divide a unit ball into five pieces and rearrange the pieces into TWO solid unit balls. (The pieces are very complicated, too complicated to have well-defined volumes.)

**Old Riddle** (Jacob Sturm). What is greater than God, more evil than the devil, the poor have it, the rich want it, and if you eat it you die?

**Answer** (Timur Dogan, Walt Wright). "Nothing."

**New Riddle** (Walt Wright). What's next in the series 77 49 36 18 ?

**New Challenge.** Winter arrives today (Thursday, December 21, 2000) at 8:37 am Eastern time. What would days and seasons be like on a planet shaped like a cube instead of a sphere?

Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for* Flatland *and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.