Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences...

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## 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...## Kakeya Needle Problem

What is the smallest-area convex set in the plane inside which a needle (unit straight line segment) can be reversed...

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...## Descartes’ Rule of Signs

Given a polynomial such as: x4 + 7×3 – 4×2 – x – 7 it is possible to say anything about how many positive real...

Continue reading...## Repunit Fun

A repunit is a whole number consisting of only 1’s, such as 1, 111, or 111111111. (It’s like a repeating unit.) These numbers have...

Continue reading...## Sine of (1/55555….)

Take your calculator, and enter a number consisting of several 5’s, such as 5555555. Now take the reciprocal of this number. Now take the...

Continue reading...## Divisibility by Eleven

It is easy to tell that the following are multiples of 11: 22, 33, 44, 55, etc. But how about:...

Continue reading...## Behold! the Pythagorean Theorem

Figure 1 shows one of the simplest proofs of the Pythagorean Theorem. It is also perhaps the earliest recorded proof, known...

Continue reading...## Why Does 0.999… = 1?

Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of...

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