Science & Math

Math Notes

27 × 81 = 2187
35 × 41 = 1435

The Lottery Paradox

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?

Divide and Conquer

divide and conquer

The Two-Envelope Paradox

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?

Universally Chilly

-40° Celsius = -40° Fahrenheit

Math Notes

407 = 43 + 03 + 73

Math Made Easy

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.

A Novel Proof

hayes trapezoid

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

hayes trapezoid area

The area of each green triangle is

hayes small triangles

And the yellow triangle is

hayes large triangle


hayes algebra

The Voting Paradox

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.

Math Notes

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.

Bertrand’s Postulate

Think of a number greater than 1. Double it.

Between these two values is at least one prime number.

Who’s in Charge Here?

“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

Math Notes

27 – 1 = 127

Right Side Up

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”:

Applied Math

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).

Math Notes

(3 + 4)3 = 343

Math Notes

9 + 10 + 11 + 12 = 13 + 14 + 15

Good Advice

“There are two rules for success,” says Raymond Smullyan. “Rule number one: Never tell all you know.”


pi equals 3

Math Notes

math notes

Double Duty

multi-magic square

A “multi-magic” square: Each row and column sums to 260; square each term and they sum to 11,180.

Half Right

“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

Simple Enough

Before his students arrived for a graduate course in logic, Raymond Smullyan wrote on the blackboard: