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