Science & Math

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.

When in Rome …

An oyster oddity: In 1954, Northwestern University biologist Frank A. Brown collected 15 oysters from the Connecticut shore and shipped them by train to Evanston, Ill. There he put them in a temperature-controlled tank in a dark room and observed them for 46 days.

The oysters opened their shells twice a day, presumably for feeding, at the time of the high tide in their home beds in Long Island Sound. After two weeks, though, their timing shifted to follow the local tides in Evanston.

Apparently they had recalibrated using the moon.


Henry is driving in the countryside with his son. For the boy’s edification, Henry identifies various objects on the landscape as they come into view. ‘That’s a cow,’ says Henry. ‘That’s a tractor,’ ‘That’s a silo,’ ‘That’s a barn,’ etc. Henry has no doubt about the identity of these objects; in particular, he has no doubt that the last-mentioned object is a barn, which indeed it is. Each of the identified objects has features characteristic of its type. Moreover, each object is fully in view, Henry has excellent eyesight, and he has enough time to look at them reasonably carefully, since there is little traffic to distract him.

Given this information, would we say that Henry knows that the object is a barn? Most of us would have little hesitation in saying this, so long as we were not in a certain philosophical frame of mind. Contrast our inclination here with the inclination we would have if we were given some additional information. Suppose we are told that, unknown to Henry, the district he has just entered is full of papier-mâché facsimiles of barns. These facsimiles look from the road exactly like barns, but are really just façades, without back walls or interiors, quite incapable of being used as barns. They are so cleverly constructed that travelers invariably mistake them for barns. Having just entered the district, Henry has not encountered any facsimiles; the object he sees is a genuine barn. But if the object on that site were a facsimile, Henry would mistake it for a barn. Given this new information, we would be strongly inclined to withdraw the claim that Henry knows the object is a barn. How is this change in our assessment to be explained?

— Alvin I. Goldman, “Discrimination and Perceptual Knowledge,” Journal of Philosophy, November 1976

Cost Cutting

“I am fond of the businessman’s paradox due to Lisa Collier: The president of a certain company offered a reward of $100 to any employee who could offer a suggestion which would save the company money. One employee suggested: ‘Eliminate the reward.'” — Raymond Smullyan

The St. Petersburg Paradox

Let’s play a game. You’ll flip a coin, and if it comes up heads I’ll give you $1. If you flip heads again I’ll give you $2, then $4, then $8, and so on. When the coin comes up tails, the game is over and you can keep your winnings.

Because I’m taking a risk, I ought to charge you an entrance fee. What’s a fair fee? Surprisingly, it seems I should charge you an infinite amount of money. With each new flip your chance of success is 1/2 but your prospective earnings double, so your total expected earnings — the earnings times their chance of being realized — is infinite:

E = (1/2 × 1) + (1/4 × 2) + (1/8 × 4) + … = ∞

Nicholas Bernoulli first described this problem in 1713. One proposed resolution is that it ignores psychology — we’re considering the monetary value of the prize rather than its value to us. Gold shines more brightly for a beggar than for a billionaire; once we’ve amassed a certain sum, the appeal of greater riches begins to diminish. “The mathematicians estimate money in proportion to its quantity,” wrote Gabriel Cramer, “and men of good sense in proportion to the usage that they may make of it.”

(Thanks, Ross.)

A Side Matter

Inscribe a pentagon in a unit circle:

Now AB × AC × AD × AE = 5.

Pleasingly, this works with any regular n-gon: A hexagon yields 6, a heptagon 7, and so on. I think it was first noted by John Bull.

The Paradox of the Divided Stick

Take a whole stick and cut it in half. Half a minute later, cut each half in half. A quarter of a minute after that, cut each quarter in half, and so on ad infinitum.

What will remain at the end of a minute? An infinite number of infinitely thin pieces? Writes Oxford philosopher A.W. Moore, “Do we so much as understand this?”

Does each piece have any width? If so, couldn’t we reassemble them to form an infinitely long stick? If not, how can we assemble them to form anything at all?

All the Way Down

Caltech number theorist Tom Apostol devised this elegant proof of the irrationality of the square root of 2.

Suppose the number is rational. Then there must be an isosceles right triangle with minimum integer sides (here, triangle ABC with sides n and hypotenuse m).

By drawing two arcs as shown, we can immediately establish triangle FDC — a smaller isosceles right triangle with integer sides.

This leads to an infinite descent. Hence n and m can’t both be integers, and the square root of 2 is irrational.

The Wason Card Task

You’re presented with the four cards above. Each has a number on one side and a color on the other. Which card(s) must be turned over to test the idea that if a card shows an even number on one face, then its opposite face is red?

In a 1966 study by Peter Wason, fewer than 10 percent of respondents correctly indicated the 8 and brown cards.

Interestingly, respondents perform significantly better when they’re presented with the same task in the context of policing a social rule (e.g., the rule is “If you are drinking alcohol then you must be over 21″ and the cards are marked “27,” “16,” “drinking Coke,” “drinking beer”). About 90 percent of people perform this task correctly — supporting the idea that our facility for such tasks evolved to catch cheaters in a social environment.

Math Notes

123456789 = ((86 + 2 × 7)5 – 91) / 34
987654321 = (8 × (97 + 6/2)5 + 1) / 34
14459929 = 17 + 47 + 47 + 57 + 97 + 97 + 27 + 97
595968 = 45 + 49 + 45 + 49 + 46 + 48
397612 = 32 + 91 + 76 + 67 + 19 + 23

36428594490313158783584452532870892261556 = 342 + 642 + 442 + 242 + 842 + 542 + 942 + 442 + 442 + 942 + 042 + 342 + 142 + 342 + 142 + 542 + 842 + 742 + 842 + 342 + 542 + 842 + 442 + 442 + 542 + 242 + 542 + 342 + 242 + 842 + 742 + 042 + 842 + 942 + 242 + 242 + 642 + 142 + 542 + 542 + 642

String Theory

A boy at the edge of a pond pulls a toy boat ashore. If he pulls in one yard of string, will the boat advance by more or less than one yard?

Surprisingly (to me), it will cover more than one yard. Because the boy is above the level of the water, he won’t pull in the entire length of string — length c will remain when the boat reaches shore. The length he’ll pull in, then, is ac.

In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side, so b + c > a and b > ac — so the boat travels a greater distance than the length of string pulled in.

A Better Invention

farrell mousetrap magic cube

This magic word cube was devised by Jeremiah Farrell. Each cell contains a unique three-letter English word, and when the three layers are stacked, the words in each row and column can be anagrammed to spell MOUSETRAP.

Setting O=0, A=1, U=2, M=0, R=3, S=6, P=0, E=9, and T=18 produces a numerical magic cube (for example, MAE = 0 + 1 + 9 = 10).

Cretan Trouble

Epimenides, a Cretan, says that all Cretans are liars. Is this a paradox? Not really: If we suppose that the statement is true then we’re led to a contradiction, but we can consistently suppose it to be false.

But, A.N. Prior writes, “We thus reach the peculiar conclusion that if any Cretan does assert that nothing asserted by a Cretan is true, then this cannot possibly be the only assertion made by a Cretan — there must also be, beside this false Cretan assertion, some true one. Yet how can there be a logical impossibility in supposing that some Cretan asserts that no Cretan ever says anything true, and that this is the only assertion ever made by a Cretan?”

Alonzo Church first raised this point in 1946: “Without factual information about other statements by Cretans, it has been proved by pure logic (so it seems) that some other statement by a Cretan, not the famous statement of Epimenides, must once have been true.”

The paradox, Prior writes, is that “such examination makes it seem possible to settle an empirical question on logical grounds.”


balance diagram

Draw two circles with three tangents as shown.

AB will always equal CD.

Skinner on Campus

A psychologist at a girl’s college asked the members of his class to compliment any girl wearing red. Within a week the cafeteria was a blaze of red. None of the girls were aware of being influenced, although they did notice that the atmosphere was more friendly. A class at the University of Minnesota is reported to have conditioned their psychology professor a week after he told them about learning without awareness. Every time he moved toward the right side of the room, they paid more attention and laughed more uproariously at his jokes, until apparently they were able to condition him right out the door.

— W. Lambert Gardiner, Psychology: A Story of a Search, 1970

The Gettier Problem

Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition:

(d) Jones is the man who will get the job, and Jones has ten coins in his pocket.

Smith’s evidence for (d) might be that the president of the company assured him that Jones would in the end be selected, and that he, Smith, had counted the coins in Jones’s pocket ten minutes ago. Proposition (d) entails:

(e) The man who will get the job has ten coins in his pocket.

Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the grounds of (d), for which he has strong evidence. In this case, Smith is clearly justified in believing that (e) is true.

But imagine, further, that unknown to Smith, he himself, not Jones, will get the job. And, also, unknown to Smith, he himself has ten coins in his pocket. Proposition (e) is then true, though proposition (d), from which Smith inferred (e), is false.

In our example, then, all of the following are true: (i) (e) is true, (ii) Smith believes that (e) is true, and (iii) Smith is justified in believing that (e) is true. But it is equally clear that Smith does not know that (e) is true; for (e) is true in virtue of the number of coins in Smith’s pocket, while Smith does not know how many coins are in Smith’s pocket, and bases his belief in (e) on a count of the coins in Jones’s pocket, whom he falsely believes to be the man who will get the job. [Does Smith know that the man who will get the job has ten coins in his pocket?]

— Edmund L. Gettier, “Is Justified True Belief Knowledge?”, Analysis, 1963

A Logic Oddity

Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason:

— If a Republican wins the election, then if it’s not Reagan who wins it will be Anderson.
— A Republican will win the election.

Yet they did not have reason to believe

— If it’s not Reagan who wins, it will be Anderson.

— Vann McGee, “A Counterexample to Modus Ponens,” Journal of Philosophy, September 1985