The Absent-Minded Driver’s Paradox

absent-minded driver's paradox

A driver is sitting in a pub planning his trip home. In order to get there he must take the highway and get off at the second exit. Unfortunately, the two exits look the same. If he mistakenly takes the first exit he’ll have to drive on a very hazardous road, and if he misses both exits then he’ll reach the end of the highway and have to spend the night at a hotel. Assign the payoff values shown above: 4 for getting home, 1 for reaching the hotel, and 0 for taking the first exit.

The man knows that he’s very absent-minded — when he reaches an intersection, he can’t tell whether it’s the first or the second intersection, and he can’t remember how many exits he’s passed. So he decides to make a plan now, in the pub, and follow it on the way home. This amounts to choosing between two policies: Exit when you reach an intersection, or continue. The exiting policy will lead him to the hazardous road, with a payoff of 0, and continuing will lead him to the hotel, with a payoff of 1, so he chooses the second policy.

This seems optimal. But then, on the road, he finds himself approaching an intersection and reflects: This is either the first or the second intersection, each with probability 1/2. If he were to exit now, the expected payoff would be

\displaystyle E = \frac{1}{2}\left ( 0 \right ) + \frac{1}{2}\left ( 4 \right ) = 2.

That’s twice the payoff of going straight! “There appear to be two contradictory optimal strategies, one at the planning stage and one at the action stage while driving,” writes Leonard M. Wapner in Unexpected Expectations. “At the pub, during the planning stage, it appears the driver should never exit. But once this plan is in place and he arrives at an exit, a recalculation with no new significant information shows that exiting yields twice the expectation of going straight.” What is the answer?

(Michele Piccione and Ariel Rubinstein, “On the Interpretation of Decision Problems with Imperfect Recall,” Games and Economic Behavior 20 [1997], 3-24.)

| Science & Math

A Little Help

In 1987, a Palermo physicist named Stronzo Bestiale published major papers in the Journal of Statistical Physics, the Journal of Chemical Physics, and the proceedings of a meeting of the American Physical Society in Monterey.

Why is this remarkable? Stronzo bestiale is Italian for “total asshole.”

Italian journalist Vito Tartamella wrote to one of “Bestiale’s” co-authors, Lawrence Livermore physicist William G. Hoover, to get the story. Hoover had been developing a sophisticated new computational technique, non-equilibrium molecular dynamics, with Italian physicist Giovanni Ciccotti. He found that the journals he approached refused to publish his papers — the ideas they contained were too innovative. But:

While I was traveling on a flight to Paris, next to me were two Italian women who spoke among themselves, saying continually: ‘Che stronzo (what an asshole)!’, ‘Stronzo bestiale (total asshole)’. Those phrases had stuck in my mind. So, during a CECAM meeting, I asked Ciccotti what they meant. When he explained it to me, I thought that Stronzo Bestiale would have been the perfect co-author for a refused publication. So I decided to submit my papers again, simply by changing the title and adding the name of that author. And the researches were published.

Renato Angelo Ricci, president of the Italian Physical Society, called the joke “an offense to the entire Italian scientific community.” But Hoover had learned a lesson: He thanked “Bestiale” at the end of another 1987 paper, saying that discussions with him had been “particularly useful.”

(From Parolacce, via Language Log. Thanks, Daniel.)

| Language · Science & Math

Four-for-All

https://commons.wikimedia.org/wiki/File:Zambezi_River_at_junction_of_Namibia,_Zambia,_Zimbabwe_%26_Botswana.jpg
Image: Wikimedia Commons

Here’s an odd bit of African geography: The finger of land in the upper left is Namibia, the region at the top is Zambia, Zimbabwe is at bottom right, and Botswana is at bottom left. Is the border between Zambia and Botswana long enough to permit a bridge to be built between the two? Or do the two peninsulas intrude far enough to make this impossible?

The answer isn’t clear. In 1970 Namibia had insisted that the four nations meet at a single point, meaning that the Kazungula Ferry linking Botswana and Zambia was illegal, as the border between them had no breadth. After an armed confrontation the ferry was sunk. Thirty-five years later Botswana and Zambia proposed building a bridge where the ferry had run. Is that geometrically permissible? The shaky consensus is that the two nations share a brief boundary of 150 meters between two “tripoints.” But the truth is as murky as the Zambezi itself.

(See Point of Interest. Thanks, Steve.)

| Oddities

Black and White

dawson chess puzzle

A curious chess puzzle by T.R. Dawson. White is to mate in four moves, with the stipulation that white men that are guarded may not move.

This seems immediately impossible. The two rooks guard one another, and one of them guards the king. And moving the pawn will place it under the king’s protection, leaving White completely frozen. How can he proceed?

Click for Answer
| Puzzles

Eodermdromes

A spelling net is the pattern made when one writes down one instance of each unique letter that appears in a word and then connects these letters with lines, spelling out the word. For instance, the spelling net for VIVID is made by writing down the letters V, I, and D and drawing a line from V to I, I to V, V to I, and I to D.

Different words produce different spelling nets, of course, but every spelling net is an example of a graph, a collection of points connected by lines. A graph is said to be non-planar if some of the lines must cross; in the case of the spelling net, this means that no matter how we arrange the letters on the page, when we connect them in order we find that at least two of the lines must cross.

A word with a non-planar spelling net is called an eodermdrome, an ungainly name that itself illustrates the idea. The unique letters in EODERMDROME are E, O, D, R, and M. Write these down and run a pen among them, spelling out the word. You’ll find that no matter how the letters are arranged, it’s never possible to complete the task without at least two of the lines crossing:

https://commons.wikimedia.org/wiki/File:Eodermdrome.png
Image: Wikimedia Commons

Ross Eckler sought all the eodermdromes in Webster’s second and third editions; another example he found is SUPERSATURATES:

https://commons.wikimedia.org/wiki/File:Supersaturates2.png
Image: Wikimedia Commons

Since spelling nets are graphs, they can be studied with the tools of graph theory, the mathematical study of such networks. One result from that discipline says that a graph is non-planar if and only if it can be reduced to one of the two patterns marked K5 and K(3, 3) above. Since both EODERMDROME and SUPERSATURATES contain these forbidden graphs, both are non-planar.

A good article describing recreational eodermdrome hunting, by computer scientists Gary S. Bloom, John W. Kennedy, and Peter J. Wexler, is here. One warning: They note that, with some linguistic flexibility, the word eodermdrome can be interpreted to mean “a course on which to go to be made miserable.”

| Language · Science & Math

The Oddfather

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Vincent Gigante, head of the Genovese crime family from 1981 to 2005, feigned mental illness for 30 years in order to throw law enforcement authorities off his trail. Beginning in the 1960s he could regularly be seen shuffling around his Greenwich Village neighborhood in pajamas, a bathrobe, and slippers, mumbling to himself, and quietly playing pinochle at a local club. His lawyers and relatives insisted he had become mentally disabled, with an IQ of 69 to 72.

But informants told the FBI that during this time he was really leading the wealthiest and most powerful crime family in the nation and a dominant force in the New York mob.

At arraignments he appeared in pajamas, and psychiatrists testified that he had been confined 28 times for hallucinations and “dementia rooted in organic brain damage.” “He was probably the most clever organized-crime figure I have ever seen,” former FBI supervisor John S. Pritchard told the New York Times. Mob rival John Gotti called him “crazy like a fox.”

It wasn’t until April 2003, in exchange for a plea deal, that he acknowledged that the whole thing had been a con to delay his racketeering trial. His lawyer said, “I think you get to a point in life — I think everyone does — where you become too old and too sick and too tired to fight.” He died in prison in 2005.

| Crime · Oddities

The Simson Line

https://commons.wikimedia.org/wiki/File:Pedal_Line.svg

The three corners of any triangle ABC define a circle that surrounds it, called its circumcircle. And for any point P on this circle, the three points closest to P on lines AB, AC, and BC are collinear.

The converse is also true: Given a point P and three lines no two of which are parallel, if the closest points to P on each of the lines are collinear, then P lies on the circumcircle of the triangle formed by the lines.

This discovery is named for Robert Simson, though, as often happens, it was first published by someone else — William Wallace in 1797.

| Science & Math

Podcast Episode 118: The Restless Corpse of Elmer McCurdy

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In 1976 a television crew discovered a mummified corpse in a California funhouse. Unbelievably, an investigation revealed that it belonged to an Oklahoma outlaw who had been shot by sheriff’s deputies in 1911 and whose remains had been traveling the country ever since. In this week’s episode of the Futility Closet podcast we’ll trace the postmortem odyssey of Elmer McCurdy, “the bandit who wouldn’t give up.”

We’ll also reflect on a Dutch artist’s disappearance and puzzle over some mysterious hospital deaths.

See full show notes …

| Death · Oddities · Podcast

Enochian

https://commons.wikimedia.org/wiki/File:Enochian_alphabet.png
Image: Wikimedia Commons

In 1581, using a “shewstone,” or crystal, astrologer John Dee and seer Edward Kelley set out to discover knowledge that couldn’t be gleaned from books or experimentation.

They succeeded: Angels gave them the “Celestial Speech,” the language that God used to create the world and taught to Adam, who lost it in his fall from paradise.

After transmitting the 21-letter alphabet above, Dee said, the angels sent him a series of texts, some with translations, that formed the basis for a vocabulary.

Some features of “Enochian” suggest that Dee was “speaking in tongues” while transcribing the language, while others show suspicious similarities to English grammar and syntax. But then, Dee maintained that modern languages arose through Adam’s attempts to reconstruct the language he had lost.

“No language has a stranger history than the Enochian language,” wrote Australian linguist Donald Laycock, who studied the curious system. “Perhaps strangest of all is that we still do not know whether it is a natural language or an invented language — or whether it is, perhaps, the language of the angels, as its originators believed.”

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