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...
Continue reading...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...
Continue reading...Arrow’s Impossibility Theorem
Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences...
Continue reading...All Numbers are Interesting
There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first...
Continue reading...Two Envelopes Paradox
I have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much...
Continue reading...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...
Continue reading...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...
Continue reading...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...
Continue reading...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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