Science & Math

A Physics Problem

You’re in a rowboat in a swimming pool, and you’re holding a cannonball. If you throw the ball into the pool, will the water level rise or fall?

Click for Answer

King Size

In 1989, paleontologists discovered the fragmentary remains of an enormous dinosaur in southern India.

If estimates are accurate, Bruhathkayosaurus was 145 feet long and weighed 240 tons.

The largest modern whale is 110 feet long and weighs 195 tons.

Math Notes

32 + 42 = 52

212 + 222 + 232 + 242 = 252 + 262 + 272

Bike Trip

The first LSD trip took place on April 19, 1943, when Swiss chemist Albert Hofmann ingested 250 micrograms and tried to go home:

I had to struggle to speak intelligibly. I asked my laboratory assistant, who was informed of the self-experiment, to escort me home. We went by bicycle, no automobile being available because of wartime restrictions on their use. On the way home, my condition began to assume threatening forms. Everything in my field of vision wavered and was distorted as if seen in a curved mirror. I also had the sensation of being unable to move from the spot. Nevertheless, my assistant later told me we had traveled very rapidly.

It’s remembered as “Bicycle Day.”

Math Notes

135 = 1 + 32 + 53
175 = 1 + 72 + 53
518 = 5 + 12 + 83
598 = 5 + 92 + 83


Once upon a time, there lived a rich farmer who had 30 children, 15 by his first wife who was dead, and 15 by his second wife. The latter woman was eager that her eldest son should inherit the property. Accordingly one day she said to him, “Dear Husband, you are getting old. We ought to settle who shall be your heir. Let us arrange our 30 children in circle, and counting from one of them, remove every tenth child until there remains but one, who shall succeed to your estate.”

The proposal seemed reasonable. As the process of selection went on, the farmer grew more and more astonished as he noticed that the first 14 to disappear were children by his first wife, and he observed that the next to go would be the last remaining member of that family. So he suggested that they should see what would happen if they began to count backwards from this lad. She, forced to make an immediate decision, and reflecting that the odds were now 15 to 1 in favour of her family, readily assented. Who became the heir?

16 children

– W.W. Rouse Ball, Mathematical Recreations & Essays, 1892

Shades of Gray

An optical illusion. Squares A and B are the same color.

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.