Science & Math

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:


Math Notes

division cancellation

Galton’s Paradox

Suppose you flip three fair coins.

Necessarily two will match, and it’s an even chance whether the third will be head or tail.

Therefore the chance that all three will match is 1/2.

Math Notes

8589934592 × 116415321826934814453125 = 1000000000000000000000000000000000


What’s the difference between six dozen dozen and half a dozen dozen?

If you answered “nothing,” reconsider.


Syllogisms offered in Lewis Carroll’s 1896 textbook in symbolic logic:

1. Babies are illogical.
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.
Therefore babies cannot manage crocodiles.

1. No interesting poems are unpopular among people of real taste.
2. No modern poetry is free from affectation.
3. All your poems are on the subject of soap bubbles.
4. No affected poetry is popular among people of taste.
5. Only a modern poem would be on the subject of soap bubbles.
Therefore all your poems are uninteresting.


Raymond Smullyan proposes a scene in which two men are regarding a blackboard. On the board is written ONLY AN IDIOT WOULD BELIEVE THIS SENTENCE.

The first man says, “Do you believe that sentence?”

The second says, “Of course not. Only an idiot would believe that sentence.”

“He clearly does believe it, yet he says he doesn’t believe it,” Smullyan says. “So he’s in the curious position of believing something and also believing that he doesn’t believe it.”

Double Duty

doubly magic square

A doubly magic square: Every row and column adds to 840 and multiplies to 2,058,068,231,856,000.