Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,...
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Domino and Square Tilings
How many ways can you tile a (1 x n) board with (1 x 1) squares and (1 x 2)...
Continue reading...Fourier Ears Only
Did you know that every sufficiently smooth function on an interval can be expressed as an infinite sum of sines...
Continue reading...Eccentricity of Conics
To each conic section (ellipse, parabola, hyperbola) there is a number called the eccentricity that uniquely characterizes the shape of the curve. A circle...
Continue reading...Volume of a Cone in N Dimensions
One of the first geometric formulas we learn in plane geometry is that the area of a triangle is: Area of a...
Continue reading...Vickrey Auction
Suppose you are hosting a silent auction to sell your antique car. The rules are: (1) prospective buyers bid for...
Continue reading...Brouwer Fixed Point Theorem
One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem....
Continue reading...Monty Hall Problem
Here’s a problem that makes the round every few years, and each time, it is hotly debated. You are on...
Continue reading...Deal or No Deal
On the TV show “Deal or No Deal”, a contestant is faced with a number of briefcases (26 in the...
Continue reading...Sum of Prime Reciprocals
It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges. But what about...
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