Non-Intuitive Mathematics

 Solution to Russel's Paradox

An analysis of the paradoxes to be avoided shows that they all result from a kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that “all propositions are either true or false.” It would seem, however, that such a statement could not be legitimate unles...

 Riddle of the Lily Pads

There is great truth in the oft-quoted riddle of the lily pads. A pond (a lake, an ocean, all apply) starts with a single lily pad. Each pad doubles per day; the pond will be full in thirty days. When is the pond only half full? On the twenty-ninth day. After the next day, the thirtieth, further growth is so fast it will, if somehow continued, overwhelm the pond and everything in it in a matter of hours.

 Positive Bias in the 2-4-6 Task

The boy's expression grew more intense. "This is a game based on a famous experiment called the 2-4-6 task, and this is how it works. I have a rule - known to me, but not to you - which fits some triplets of three numbers, but not others. 2-4-6 is one example of a triplet which fits the rule. In fact... let me write down the rule, just so you know it's a fixed rule, and fold it up and give it to you. Please don't look, since I infer from earlier that you can read upside-down." The boy said ...

 The Monty Hall Problem

Here’s how Monty’s deal works, in the math problem, anyway. (On the real show it was a bit messier.) He shows you three closed doors, with a car behind one and a goat behind each of the others. If you open the one with the car, you win it. You start by picking a door, but before it’s opened Monty will always open another door to reveal a goat. Then he’ll let you open either remaining door. Suppose you start by picking Door 1, and Monty opens Door 3 to reveal a goat. Now what should y...

 1 = 0.999999999…

This simple equation, which states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one, is the favorite of mathematician Steven Strogatz of Cornell University. "I love how simple it is — everyone understands what it says — yet how provocative it is," Strogatz said. "Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity."