"... Look, there is an infinite array of numbers. Suppose you spell them out --- the whole infinite array ---"

"Can't be done," said Jennings. "How can you spell out every one of an infinite number?"

"In imagination," said Baranov impatiently. "Now arrange them all --- the whole infinite set of them --- in alphabetical order. Which number is first in line?"

Jennings said, "How can you tell unless you look at all the numbers? And how can you look at all of an infinite number?"

"Because there's a pattern to number names," said Baranov. "There may be an infinite set of numbers, but there are only a small number of ways in which their names are formed. The first number in line, alphabetically is [censored]. Nothing comes ahead of it. There's no number in the entire infinite array that starts with 'a', 'b', [censored], and how do you like that?"

"What about 'billion'?" I said.

Baranov sneered at me elaborately. "That's not a number name. If you write the number 'one' followed by nine zeroes, that's not 'billion' starting with 'b'; that's 'one billion' starting with 'o.' "

At this point, Griswold, without seeming to interrupt his soft snore, said, "And what's the last number in line?"

2. What is the answer to Griswold's question? [2]

3. Every one of the ten Arabic digits is included in the sequence 8, 5, 4, 9, 7, 6, 3, 2, 0 except for 1. Where does it rightfully belong in the sequence? [1]

4. What should the next number in the sequence 1, 2, 6, 12, 60, 420, 840, ... be? [2]

5. Suppose you draw a number of circles on a blank piece of paper. This divides up the paper into a number of regions whose borders are made up of circular arcs. You can paint these regions using only two colours in such a way that no two regions that have a common border (bigger than a single point!) have the same colour. Prove this without using induction. [4]

Department of Mathematics

Trent University