We’ve had some pretty smart presidents. James Garfield devised this proof of the Pythagorean theorem in 1876, while serving in the House of Representatives:

The area of the trapezoid above is

The area of each green triangle is

And the yellow triangle is

So:

Suppose we hold an election with three candidates, X, Y, and Z. And suppose the voters fall into three groups:

Group 1 prefers, in order, X, Y, Z
Group 2 prefers, in order, Y, Z, X
Group 3 prefers, in order, Z, X, Y

Now, if Candidate X wins, his opponents can rightly object that a majority of voters would have preferred Candidate Z. And corresponding arguments can be made against the other candidates. So even though we’ve held a fair election, it’s impossible to establish majority rule.

The Marquis de Condorcet noted this oddity in the 1700s; it’s sometimes known as Condorcet’s paradox.

3² – 2³ = 3 – 2

Think of a number greater than 1. Double it.

Between these two values is at least one prime number.

“A hen is only an egg’s way of making another egg.” — Samuel Butler

“A zygote is a gamete’s way of producing more gametes. This may be the purpose of the universe.” — Robert Heinlein

“The nucleic acids invented human beings in order to be able to reproduce themselves even on the moon.” — Sol Spiegelman

2⁷ – 1 = 127

Set a beetle or a turtle on its back and it will right itself.

Remarkably, so will the Gömböc — a mathematical shape that can’t be knocked down. Set it down in any position and it will always “get to its feet”:

Each term in the Fibonacci sequence is derived by adding the two preceding terms:

0, 1, 1, 2, 3, 5, 8, 13, 21 …

Remarkably, you can use successive terms to convert miles to kilometers:

8 miles ≈ 13 kilometers
13 miles ≈ 21 kilometers

This works because the two units stand in the golden ratio (to within 0.5 percent).

(3 + 4)³ = 343