We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as...
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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...Farmers and Pesky Birds
Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in...
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...Continuous but Nowhere Differentiable
You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have...
Continue reading...Euler’s Product Formula
Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it...
Continue reading...How many Rationals?
How many rational numbers are there? Yes, infinitely many, I hear you say. But how large is that infinity? Are...
Continue reading...Hyperbolic Geometry
In the Fun Fact on Spherical Geometry, we saw an example of a space which is curved in such a way...
Continue reading...Volume of a Ball in N Dimensions
The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 <= 1. What...
Continue reading...Sphere Eversions
If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it...
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