27 × 81 = 2187
35 × 41 = 1435

Imagine a lottery with 1,000 tickets.

It’s rational to believe that one ticket will win.

But it’s also rational to believe that the first ticket will not win—nor the second, nor the third, and so on.

And isn’t that equivalent to believing that no ticket will win?

Here are two envelopes. One contains twice as much money as the other. You must choose one, and then consider whether to keep it or exchange it for mine. Which should you do?

It would seem advantageous to switch: Depending on which envelope you started with, you’ll either lose a little or gain a lot. (If your unopened envelope contains $10, for example, the other must contain $5 or $20.)

So we trade envelopes and I offer you the same deal. But now the same reasoning applies, so it makes sense to trade again. Indeed, it seems reasonable to keep exchanging envelopes forever, without ever opening one. How can this be?

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