If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the...

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## Klein Bottle

A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;...

Continue reading...## Medical Tests and Bayes’ Theorem

Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that...

Continue reading...## Riemann Hypothesis

If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our...

Continue reading...## Dinner Party Problem

How many people must you have at dinner to ensure that there are a subset of 3 people who all...

Continue reading...## Continuum Hypothesis

We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with...

Continue reading...## Equidecomposability

Two sets A and B are said to be equidecomposable if you can partition set A into a finite number of subsets...

Continue reading...## Sum of Cubes and Beyond

We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the...

Continue reading...## Envy-free Cake Division

Say you and a friend wish to share a cake. What is a “fair” way to split it? Probably you...

Continue reading...## Odd Numbers in Pascal’s Triangle

Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N...

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