Science & Math

Healthcare Reform

The statement “You will recover from this illness” is either true or false. If it’s true, then it has been true for all eternity, and you’ll recover whether you summon a doctor or not.

If the statement is false, then it has always been false, and you will not recover even with a doctor’s aid.

So there is no point in calling a doctor.

(From Cicero’s De Fato.)

A Good Man

Should other species be regarded as human? In 1779 Lord Monboddo proposed that orangutans should: They walk upright, use weapons, form societies, build shelters, and behave with “dignity and composure.” “If … such an Animal be not a Man, I should desire to know in what the essence of a Man consists, and what it is that distinguishes a Natural Man from the Man of Art?”

Thomas Love Peacock mocked this view in his 1817 novel Melincourt, in which a civilized orangutan (“Sir Oran Hout-ton”) is elected to Parliament. And an anonymous wag objected even to the satire:

The author of a novel lately written,
Entitled “Melincourt,”
(‘Tis very sweet and short),
Seems indeed by some wondrous madness bitten,
Thinking it good
To take his hero from the wood:
And though I own there’s nothing treasonable
In making ouran-outangs reasonable,
I really do not think he should
Go quite the length that he has done,
Whether for satire or for fun,
To make this creature an M.P.
As if mankind no wiser were than he.
However, those who’ve read it
Must give the author credit
For skill and ingenuity,
Although it have this monstrous incongruity.

But today Monboddo’s view is on the ascendancy. Harvard legal scholar Steven M. Wise argues that orangutans — as well as chimpanzees, bonobos, elephants, parrots, dolphins, and gorillas — deserve legal personhood. “Ancient philosophers claimed that all nonhuman animals had been designed and placed on this earth just for human beings,” he writes. “Ancient jurists declared that law had been created just for human beings. Although philosophy and science have long since recanted, the law has not.”

Heavenly Form

The U.S. Customs Service received an odd bit of paperwork on July 24, 1969:

apollo 11 customs form

All three Apollo 11 astronauts signed the form, which was filed at the Honolulu airport on the day they splashed down.

“Yes, it’s authentic,” NASA spokesperson John Yembrick told “It was a little joke at the time.”

See Business Trip and Space Bills.

Math Notes

math notes

(Thanks, Pablo.)

Surface Matters

If you touch a gold ball, you touch its surface and you touch gold. It seems reasonable to conclude that the surface is made of gold. But University of Exeter computer scientist Antony Galton points out that the surface is two-dimensional; it can’t contain any quantity of gold.

What then is it? We can’t say it’s the outermost layer of gold atoms, for that’s a film with two surfaces. And we can’t say it’s an abstract boundary with no physical existence, for we can see it and touch it. So what is it?

J.L. Austin asked, “Where and what exactly is the surface of a cat?”

Sure Thing

[Lewis Carroll] told me of a simple, too simple, rule by which, he thought, one could be almost sure of making something at a horse-race. He had on various occasions noted down the fractions which represented the supposed chances of the competing horses, and had observed that the sum of these chances amounted to more than unity. Hence he inferred that, even in the case of such hard-headed men as the backers, the wish is often father to the thought; so that they are apt to overrate the chances of their favourites. His plan, therefore, was to bet against all the horses, keeping his own stake the same in each case. He did not pretend to know much about horse-racing, and I probably know even less; but I understand that it would be impossible to adjust the hedging with sufficient exactitude — in fact, to get bets of the right amount taken by the backers.

— Lionel Arthur Tollemache, Old and Odd Memories, 1908

Pi Coincidences

The number 360 is centered across the 360th decimal place of π:

pi coincidences - 360

6998970 = 36 + 19 + 49 + 18 + 59 + 97 + 20

6998971 = 36 + 19 + 49 + 18 + 59 + 97 + 21

(Thanks, Pablo.)

pi coincidences - approximations

The 22nd, 7th, 355th, 113th, and 52163rd digits of π (counting from the 3) are 2s.

The 16604th digit, alas, is a 1 — but it’s flanked by 2s.

After You

At the end of your visit to an elderly, infirm relative who lives alone, the relative says, ‘I’m sorry but my arthritis won’t let me get up from this chair today. You’ll have to show yourself out.’ How can you show yourself out of someone’s house? If you know the way out, you can act as a guide to someone else. But how can you act as your own guide?

— T.S. Champlin, Reflexive Paradoxes, 1988

Johnson Circles

johnson circles

If three hula hoops cover a common point, then a fourth hoop will cover their remaining intersections.

Safe Passage

Mathematician G.H. Hardy had an ongoing feud with God. Once, after spending a summer vacation in Denmark with Harald Bohr, he found he’d have to take a small boat across the tempestuous North Sea to return to England. Before boarding, he sent Bohr a postcard that said “I have proved the Riemann hypothesis. — G.H. Hardy.”

When Bohr excitedly asked about this later, “Oh, that!” Hardy said. “That was just insurance. God would never let me drown if it meant I’d get undue credit.”


  • Connecticut didn’t ratify the Bill of Rights until 1939.
  • Can one pity a fictional character?
  • 64550 = (64 – 5) × 50
  • BILLOWY is in alphabetical order, WRONGED in reverse.
  • “The essence of chess is thinking about the essence of chess.” — David Bronstein

Presto Chango

jones reversible magic square

From Samuel Isaac Jones, Mathematical Wrinkles (1929), a magic square with a twist:

“It will be observed that this square when turned upside down is still magic.”

Math Notes

hunter figure curio

Discovered by J.A.H. Hunter.

Half and Half

half and half 1

A line that bisects the right angle in a right triangle also bisects a square erected on the hypotenuse.


half and half 2

Time and Motion

If a second is defined by reference to the rotation of the earth on its axis, i.e. as 1/60 of 1/60 of 1/24 of the time between 2 identical positions of the Greenwich meridian relatively to the fixed stars, then, if the earth rotated 10 times more slowly than it does now, it would be possible to run 10 yds. in a second, instead of only a yard as now, and a second would be 10 times longer than it is now; but if cinema machines still moved as fast as they do now, it would still be quite impossible for any one to see a succession of static pictures instead of a moving one. Don’t we mean by a second the length of time which is now 1/60 of 1/60 of 1/24 of the time between etc.?

— G.E. Moore, Commonplace Book, 1962

Compound Interest

On Jan. 18, 1897, California farmer George Jones bought a quantity of livestock feed from Henry B. Stuart of San Jose. As security he signed a $100 promissory note that bore 10 percent interest per month, compounded monthly.

They had agreed that Jones would pay the debt in three months, but the note had run for almost 25 years when Stuart got tired of waiting and told his lawyer to sue. Judge J.R. Welch of the Superior Court of Santa Clara entered this judgment on March 6, 1922:

“Wherefore, by virtue of the law and the facts, it is Ordered, Adjudged and Decreed that said Plaintiff have and recover from said Defendant the sum of $304,840,332,912,685.16 with interest thereon at the rate of 7% per annum until paid, together with the further sum of $50.00 Plaintiff’s attorney’s fees herein with interest thereon at the rate of 7% per annum until paid.”

That’s $304 trillion, “more money than there is in the world, outside of Russia,” the New York Tribune reported drily. Jones paid $19.69 and filed for bankruptcy.

Divide and Conquer

In 1980, Colorado math teacher William J. O’Donnell was explaining that

divide and conquer 1

when a student noted that

divide and conquer 2

“My immediate reaction was that this student had stumbled onto a special case where this algorithm worked,” O’Donnell wrote in a letter to Mathematics Teacher. “Later, a couple of minutes of work revealed that this technique works for all fractions. Let a, b, c, and d be integers. Then

divide and conquer 3

“Whereas this method can be conveniently applied on occasion, it does not offer the student much advantage when c does not divide a and d does not divide b.”


Darth Vader is piloting a barge to Salt Lake City to give a workshop on evildoing. Suddenly he finds himself approaching a crumbling brick aqueduct, at the foot of which is a basket of adorable kittens. He struggles to stop the barge, but it’s too late. The terrified kittens mew piteously, but they’re too weak to escape. Inexorably, implacably, the barge floats out directly over the basket. What happens?

Nothing happens. The barge displaces its weight in water, so there’s no additional load on the aqueduct.

The workshop is a great success.


  • No bishop appears in Through the Looking-Glass.
  • Can a law compel us to obey the law?
  • 98415 = 98-4 × 15
  • Why does the ghost haunt Hamlet rather than Claudius?
  • “Put me down as an anti-climb Max.” — Max Beerbohm, declining to hike to the top of a Swiss Alp

The Before-Effect

Let the peal of a gong be heard in the last half of a minute, a second peal in the preceding 1/4 minute, a third peal in the 1/8 minute before that, etc. ad infinitum. … Of particular interest is the following puzzling case. Let us assume that each peal is so very loud that, upon hearing it, anyone is struck deaf — totally and permanently. At the end of the minute we shall be completely deaf (any one peal being sufficient), but we shall not have heard a single peal! For at most we could have heard only one of the peals (any single peal striking one deaf instantly), and which peal could we have heard? There simply was no first peal.

— Jose Benardete, Infinity: An Essay in Metaphysics, 1964

Crackpot Apocalypse

Various writers throughout the 19th century confidently reported that they’d found the true and exact value of π. Unfortunately, they all gave different answers. In 1977 DePauw University mathematician Underwood Dudley tried to make sense of this by compiling 50 of their pronouncements:

pi estimates - underwood dudley

He concluded that π is decreasing. The best fit is πt = 4.59183 – 0.000773t, where t is the year A.D. — it turns out we passed 3.1415926535 back in 1876 and have been heading downward ever since.

And that means trouble: “When πt is 1, the circumference of a circle will coincide with its diameter,” Dudley writes, “and thus all circles will collapse, as will all spheres (since they have circular cross-sections), in particular the earth and the sun. It will be, in fact, the end of the world, and … it will occur in 4646 A.D., on August 9, at 4 minutes and 27 seconds before 9 p.m.”

There is some good news, though: “Circumferences of circles will be particularly easy to calculate in 2059, when πt = 3.”

(Underwood Dudley, “πt,” Journal of Recreational Mathematics 9:3, March 1977, p. 178)

The Phantom Save

Andy fires a shot at the goal, but it’s deflected by his opponent Bill. If Bill had not reached the ball, it would have struck Charlie, Andy’s teammate. Roberto Casati asks, “Should Bill get credit for the save?”

He: Not quite. After all, the ball was not going to score anyway; it would have hit Charlie’s body.

She: But neither would it be right to say that anything happened thanks to Charlie. After all, Charlie did nothing.

He: But then who is responsible for spoiling Andy’s shot?

“Cases like this one are indicative of a deep conceptual tension,” Casati writes. “I am walking in the rain. My umbrella is open and I am wearing a hat, so my head is not getting wet. But why is that so? It’s not because of the umbrella, because I’m wearing my hat. And it’s not because of my hat, for I have an umbrella.”

From Casati’s excellent book Insurmountable Simplicities. See also In the Dark.

Math Notes

614,656 = 284
6 + 1 + 4 + 6 + 5 + 6 = 28

1,679,616 = 364
1 + 6 + 7 + 9 + 6 + 1 + 6 = 36

8,303,765,625 = 456
8 + 3 + 0 + 3 + 7 + 6 + 5 + 6 + 2 + 5 = 45

52,523,350,144 = 347
5 + 2 + 5 + 2 + 3 + 3 + 5 + 0 + 1 + 4 + 4 = 34

20,047,612,231,936 = 468
2 + 0 + 0 + 4 + 7 + 6 + 1 + 2 + 2 + 3 + 1 + 9 + 3 + 6 = 46

The Infected Checkerboard

the infected checkerboard

From the Soviet magazine KVANT, 1986:

On an n × n checkerboard, a square becomes “infected” if at least two of its orthogonal neighbors are infected. For example, if the main diagonal is infected (above), then the infection will spread to the adjoining diagonals and on to the whole board. Prove that the whole board cannot become infected unless there are at least n sick squares at the start.

The key is to notice that when a square is infected, at least two of its edges are absorbed into the infected area, while at most two of its edges are added to the boundary of the infection. Thus the perimeter of the infected area can’t increase; in order for the full board (with perimeter 4n) to become infected, there must be at least n infected squares to begin with.