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?
-40° Celsius = -40° Fahrenheit
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
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
Georg Wilhelm Richmann was attending a meeting at the St. Petersburg Academy of Sciences in August 1753 when he heard thunder. He ran home with another man, hoping to record how an insulated rod responded to an electrical storm.
He succeeded, in a way: A ball of lightning leapt from the rod and struck Richmann in the head, killing him instantly and knocking his companion unconscious. That makes Richmann the first person in history to die while conducting electrical experiments.
Joseph Priestley wrote, “It is not given to every electrician to die in so glorious a manner as the justly envied Richmann.” That’s one way to look at it.
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
9 + 10 + 11 + 12 = 13 + 14 + 15
“There are two rules for success,” says Raymond Smullyan. “Rule number one: Never tell all you know.”
A “multi-magic” square: Each row and column sums to 260; square each term and they sum to 11,180.
“Numero deus impare gaudet [the god delights in odd numbers].” — Virgil
“Why is it that we entertain the belief that for every purpose odd numbers are the most effectual?” — Pliny
“This is the third time; I hope good luck lies in odd numbers. … They say there is divinity in odd numbers, either in nativity, chance, or death.” — Shakespeare, The Merry Wives of Windsor
Before his students arrived for a graduate course in logic, Raymond Smullyan wrote on the blackboard:
PLEASE DO NOT ERASE — BECAUSE IF YOU DO, THOSE WHO COME LATER WON’T KNOW THAT THEY SHOULDN’T ERASE.