# paradox

## St. Petersburg Paradox

A new casino offers the following game: you toss a coin until it comes up heads. If the first heads...

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## Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...

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## Rubber Bands Stuck on a Torus

Consider a rubber tire (a torus) with a hole in it. Suppose that there is a green rubber band stuck...

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## Arrow’s Impossibility Theorem

Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences...

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## All Numbers are Interesting

There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first...

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## Two Envelopes Paradox

I have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much...

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## Gabriel’s Trumpet

The surface of revolution of y = 1/xfor x greater than 1, has finite volume but infinite surface area! Put another way, this...

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## Derivative Paradox

Here’s a fun (but untrue) fact. You know from calculus that the derivative of x2 is 2x. But what’s wrong with the following...

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## One Equals Zero: Integral Form

Consider the following integral: INTEGRAL (1/x) dx Perform integration by parts: let u = 1/x , dv = dx du = -1/x2 dx...

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## One Equals Zero!

The following is a “proof” that one equals zero. Consider two non-zero numbers x and y such that x =...

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