Science & Math

Richard’s Paradox

Clearly there are integers so huge they can’t be described in fewer than 22 syllables. Put them all in a big pile and consider the smallest one. It’s “the smallest integer that can’t be described in fewer than 22 syllables.”

That phrase has 21 syllables.

Math Notes

84 + 24 + 04 + 84 = 8208

Buffon’s Needle

Remarkably, you can estimate π by dropping needles onto a flat surface. If the surface is ruled with lines that are separated by the length of a needle, then:

buffon's needle

drops is the number of needles dropped. hits is the number of needles that touch a line. The method combines probability with trigonometry; a needle’s chance of touching a line is related to the angle at which it comes to rest. It was discovered by the French naturalist Georges-Louis Leclerc in 1777.

Clarke’s Law

Clarke’s Third Law: Any sufficiently advanced technology is indistinguishable from magic.

Benford’s Corollary: Any technology distinguishable from magic is insufficiently advanced.

Raymond’s Second Law: Any sufficiently advanced system of magic would be indistinguishable from a technology.

Sterling’s Corollary: Any sufficiently advanced garbage is indistinguishable from magic.

Langford’s application to science fiction: Any sufficiently advanced technology is indistinguishable from a completely ad-hoc plot device.

Math Notes

13 + 33 + 63 = 244
23 + 43 + 43 = 136

The Necktie Paradox

You and I are having an argument. Our wives have given us new neckties, and we’re arguing over which is more expensive.

Finally we agree to a wager. We’ll ask our wives for the prices, and whoever is wearing the more expensive tie has to give it to the other.

You think, “The odds are in my favor. If I lose the wager, I lose only the value of my tie. If I win the wager, I gain more than the value of my tie. On balance I come out ahead.”

The trouble is, I’m thinking the same thing. Are we both right?


“Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” — Paul Erdös

Math Notes


… are all prime. So are:


But see Not So Fast.

No Comment

Viagra keeps plants from wilting.

Israeli and Australian researchers found that a low concentration in water doubled the shelf life of cut flowers, from one week to two weeks.

Recursive Gratitude

Mathematician J.E. Littlewood once wrote a paper for the French journal Comptes Rendus. A Prof. M. Riesz did the translation, and at the end Littlewood found three footnotes:

I am greatly indebted to Prof. Riesz for translating the present paper.

I am indebted to Prof. Riesz for translating the preceding footnote.

I am indebted to Prof. Riesz for translating the preceding footnote.

Littlewood notes that this could have gone on indefinitely but “I stop legitimately at number 3: however little French I know I am capable of copying a French sentence.”

Math Notes

73 × 9 × 42 = 7 × 3942

All Art Is Theft

Irish astronomer William Parsons might have been surprised to see van Gogh’s The Starry Night appear in 1889.

He had drawn this sketch of the Whirlpool Galaxy 44 years earlier:

Pop Quiz

When calculating prodigy Truman Henry Safford was 10 years old, the Rev. H.W. Adams asked him to square the number 365,365,365,365,365,365 in his head. Dr. Adams wrote:

He flew around the room like a top, pulled his pantaloons over the tops of his boots, bit his hands, rolled his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony, until in not more than a minute said he, 133,491,850,208,566,925,016,658,299,941,583,255!

Safford (1836-1901) went to Harvard and became director of the Hopkins Observatory at Williams College. Strangely, his calculating abilities seemed to wane as he got older.

Math Notes

Hate exponents? Just cancel them:

canceling exponents 2

canceling exponents 1

See also Reductio Ad Absurdum.

Euler’s Identity

You know these numbers:


On the surface they appear unrelated. e is the base of natural logarithms, i is imaginary, π concerns circles. But, amazingly:

Euler's identity

Harvard mathematician Benjamin Peirce told a class, “It is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.”

Still a Sin?

A man’s likelihood of being gay increases by 33 percent for each older brother he has.


In 2004 a mysterious billboard appeared in Silicon Valley; Cambridge, Mass.; Seattle; and Austin, Texas. It read:

{first 10-digit prime found in consecutive digits of e}.com

Most people know that e (2.718281828 …) is the base of natural logarithms, but searching it for a 10-digit prime string is a considerable task — the first such string, 7427466391, starts at the 101st digit.

Solvers who went to found an even more difficult problem to solve. But solving that led them to a page at Google Labs … inviting them to submit a resume.

A Stubborn Prime

Type 120121 into a calculator and you’ll find it’s prime every way you look at it: right side up, upside down (121021), in a mirror (151051), or both (150151).

Math Notes

35 – 32 – 52 = 75 – 72 – 52


A neutron walks into a bar and orders a beer.

“How much do I owe you?” he says.

“For you,” says the bartender, “no charge.”

A Mathematical Limerick

math limerick

The integral z-squared dz
From one to the cube root of three
Times the cosine
Of three pi over nine
Equals log of the cube root of e.

Math Notes

144648 = 861 × 168 = 492 × 294

185472 = 672 × 276 = 384 × 483

9949716 = 2583 × 3852 = 1476 × 6741

16746912 = 2556 × 6552 = 4473 × 3744

The Goldbach Conjecture

Every even number is the sum of two primes.

Is that true? No one knows. Originally proposed in 1742, it’s been tested as far as 1018, but the jury’s still out.

Black Tie Optional

English biologist Richard Owen designed a collection of life-size concrete dinosaurs for London’s Crystal Palace.

On New Year’s Eve 1853, he hosted a dinner for 21 inside the iguanodon.