Science & Math

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.

Exit Strategies

The Roman senator who dies as a result of plunging a dagger into his heart commits suicide. He kills himself. But what about the twentieth-century suicide who places his head on the railway line and is crushed to death by the train he normally catches each morning to the office? Wasn’t he killed by the train? Then did he kill himself into the bargain too? Exactly what was it that killed him? What do you have to have done in order to count as having killed yourself?

— T.S. Champlin, Reflexive Paradoxes, 1988

The Arrow Paradox

At any given instant, an arrow in flight is where it is, occupying a space equal to itself. It cannot move during the instant, for that would require the instant to have parts.

This seems to mean that motion is impossible. Aristotle writes, “If everything, when it is behaving in a uniform manner, is continually either moving or at rest, but what is moving is always in the now, then the moving arrow is motionless.”

Bertrand Russell adds, “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever. … The more the difficulty is meditated, the more real it becomes.”

Achilles Recalled

A fragment from Lewis Carroll, Nov. 22, 1874:

A. And thus your favourite paradox, my dear D., is finally disproved of, and Achilles and the Tortoise will walk off hand in hand. No argument of any sort can be maintained, which would prove him not to overtake it.

D. No mathematical argument, you mean; for, if you permit me a classical one, I will contend that the Tortoise was nothing but the “Testudo” of the ancients, a machine of common use in Sieges — that it was at that moment moving against the walls of Troy — and that the true reason why Achilles did not overtake it was simply that he was sulking in his tent and never went near it.

S. I beg to limit this discussion to mathematical argument.

D. Be it so. And the mathematical argument you dispose of, as I understand you, by the assertion that we find ourselves at last among indivisible distances and indivisible periods of time, and thus you propose to plunge us, however reluctant we may be to take the leap, into the dark abyss of the Inconceivable?

S. That is my solution of the paradox.

D. Granting, for argument’s sake, that the paradox is thus finally disposed of, let me ask you a question or two. These indivisible distances — are they equal, or unequal?

S. Am I bound to choose one or other of these categories?

D. I fear I can offer you no third.

S. Well then, as I do not clearly see what you are aiming at, I will, for the present, say “unequal,” reserving to myself however the right of substituting “equal” should I see reason to do so.

D. The privilege is an unusual one, but I will not object to your exercising it. Let them then be: unequal. Now take two of these unequal distances: lay them side by side, so as to coincide at one end: will they coincide at the other end also?

S. Surely not.

D. There will therefore be a difference between them: and this difference, being homogeneous with the things differing, will itself be a distance?

S. I cannot deny it.

D. Divisible, shall we say? Or indivisible?

S. (laughing) Indivisible, of course. You would not wish me to imagine a divisible distance less than an indivisible one?

D. You shall please yourself in that matter. Let me now add together these two lesser indivisible distances. Will their sum total be divisible or indivisible, think you?

S. (after a pause) It occurs to me that I would rather take the other horn of your dilemma, and say that these indivisible distances are all equal.

D. With all my heart. They shall now be all equal. And we will suppose that Achilles has just passed over one of the indivisible distances. What time would you say that he occupied in doing so?

S. An indivisible time, clearly.

D. But the Tortoise had previously passed over the same indivisible distance: how long do you suppose he took to do it?

S. As he travelled at only half the pace of Achilles, it is evident that he required two of our indivisible periods of time.

D. No doubt. But now tell me — at the end of the first of these indivisible periods of time, where had the Tortoise got to?

S. I will trouble you to pass the wine. I think I should like another half-glass of sherry.

Hidden Depths

Image: Flickr

It is familiarly said that beer … is an acquired taste; one gradually trains oneself — or just comes — to enjoy that flavor. What flavor? The flavor of the first sip? No one could like that flavor, an experienced beer drinker might retort: ‘Beer tastes different to the experienced beer drinker. If beer went on tasting to me the way the first sip tasted, I would never have gone on drinking beer! Or to put the same point the other way around, if my first sip of beer had tasted to me the way my most recent sip just tasted, I would never have had to acquire the taste in the first place! I would have loved the first sip as much as the one I just enjoyed.’ If we let this speech pass, we must admit that beer is not an acquired taste. No one comes to enjoy the way the first sip tasted. Instead, prolonged beer drinking leads people to experience a taste they enjoy, but precisely their enjoying the taste guarantees that it is not the taste they first experienced.

— Daniel Dennett, “Quining Qualia,” from Consciousness in Contemporary Science, 1988

The Paradox of Self-Deception

If ever a person A deceives a person B into believing that something, p, is true, A knows or truly believes that p is false while causing B to believe that p is true. So when A deceives A (i.e., himself) into believing that p is true, he knows or truly believes that p is false while causing himself to believe that p is true. Thus, A must simultaneously believe that p is false and believe that p is true. But how is this possible?

— Alfred R. Mele, “Two Paradoxes of Self-Deception,” in Self-Deception and Paradoxes of Rationality, 1998

Curious Company

In his autobiography, mathematician Norbert Wiener describes three particular dons he came to know at Trinity College, Cambridge, in 1914:

“It is impossible to describe Bertrand Russell except by saying that he looks like the Mad Hatter. … [J.M.E.] McTaggart … with his pudgy hands, his innocent, sleepy air, and his sidelong walk, could only be the Dormouse. The third, G.E. Moore, was a perfect March Hare. His gown was always covered with chalk, his cap was in rags or missing, and his hair was a tangle which had never known the brush within man’s memory.”

The three together became known as the Mad Tea Party of Trinity. Though they appeared 50 years after their fictional counterparts, Wiener wrote, “the caricature of Tenniel almost argues an anticipation on the part of the artist.”

Plane Truth

Excerpts from One Hundred Proofs That the Earth Is Not a Globe, a pamphlet distributed by William Carpenter in 1885:

  • “If the Earth were a globe, rolling and dashing through ‘space’ at the rate of ‘a hundred miles in five seconds of time,’ the waters of seas and oceans could not, by any known law, be kept on its surface — the assertion that they could be retained under these circumstances being an outrage upon human understanding and credulity.”
  • “Astronomers tell us that, in consequence of the Earth’s ‘rotundity,’ the perpendicular walls of buildings are, nowhere, parallel, and that even the walls of houses on opposite sides of a street are not! But, since all observation fails to find any evidence of this want of parallelism which theory demands, the idea must be renounced as being absurd and in opposition to all well-known facts.”
  • “If we examine a true picture of the distant horizon, or the thing itself, we shall find that it coincides exactly with a perfectly straight and level line.”
  • “The Newtonian theory of astronomy requires that the Moon ‘borrow’ her light from the Sun. Now, since the Sun’s rays are hot and the Moon’s light sends with it no heat at all, it follows that the Sun and Moon are ‘two great lights,’ as we somewhere read, [and] that the Newtonian theory is a mistake.”
  • “If a projectile be fired from a rapidly moving body in an opposite direction to that in which the body is going, it will fall short of the distance at which it would reach the ground if fired in the direction of motion. Now, since the Earth is said to move at the rate of nineteen miles in a second of time, ‘from west to east,’ it would make all the difference imaginable if the gun were fired in an opposite direction. But … there is not the slightest difference, whichever way the thing may be done.”

Staunch flat-earther Wilbur Glenn Voliva (1870-1942) asked: “Where is the man who believes he can jump into the air, remaining off the earth one second, and come down to earth 193.7 miles from where he jumped up?” Hard to argue with that.