Welcome to Week 12 of the Wednesday Morning Math Challenge. You can find solutions to the Week 11 puzzles here, along with — more importantly — discussions of some ways to approach them. Remember that the real goal here is to think creatively.

This week, we’re going to look at one problem — but it’s a hard one.

Consider the set of integers: 0, ±1, ±2, ±3, ±4, …

Even though it’s infinite, this set is considered countable, because we can count them. You could do it on your fingers! You’d have to go on forever, but you could do it without skipping anything.

Now consider the set of numbers between 0 and 1. Are those numbers countable? How do you know?