407 = 4³ + 0³ + 7³

142857 is a cyclic number — you can find its multiples simply by rotating its digits:

• 142857 × 1 = 142857
• 142857 × 2 = 285714
• 142857 × 3 = 428571
• 142857 × 4 = 571428
• 142857 × 5 = 714285
• 142857 × 6 = 857142

Also: 142857² = 20408122499, and 20408 + 122449 = 142857.

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”: