Consider an infinite “tower of powers” of x, defined by x^x^x^… = x^(x^(x^…)) Can we find a value of x...
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High-Dimensional Spheres in Cubes
How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle...
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...Koch Snowflake
Snowflakes are amazing creations of nature. They seem to have intricate detail no matter how closely you look at them....
Continue reading...Spherical Pythagorean Theorem
Did you know there is a version of the Pythagorean Theorem for right triangles on spheres? First, let’s define precisely what we...
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...Products of Sums of Two Squares
Here’s a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the...
Continue reading...Thinned-Out Harmonic Series
You’re probably already aware that the harmonic series, which is the sum of the reciprocals of all natural numbers, diverges. In...
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...Dominoes on a Chessboard
A standard 8×8 chessboard can easily be covered (tiled) with non-overlapping dominoes (1×2 pieces): simply use 4 dominoes in each...
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