Building Schemes

In 1983, East Carolina University mathematicians Thomas Chenier and Cathy Vanderford programmed a computer to find the best strategies in playing Monopoly. The program kept track of each players’ assets and property, and subroutines managed the decisions whether to buy or mortgage property and the results of drawing of Chance and Community Chest cards. They auditioned four basic strategies (I think all of these were in simulated two-player games):

Bargain Basement. Buy all the unowned property that you can afford, hoping to prevent your opponent from gaining a monopoly.
Two Corners. Buy property between Pennsylvania Railroad and Go to Jail (orange, red, and yellow), hoping your opponent will be forced to land on one on each trip around the board.
Controlled Growth. Buy property whenever you have $500 and the color group in question has not yet been split by the two players. Hopefully this will allow you to grow but retain enough capital to develop a monopoly once you’ve acquired one.
Modified Two Corners. This is the same as Two Corners except that you also buy the Boardwalk-Park Place group.

After 200 simulated games, the winner was Controlled Growth, with 88 wins, 79 losses, and 33 draws. Bargain Basement players tended to lack money to build houses, and Two Corners gave the opponent too many opportunities to build a monopoly and was vulnerable to interference by the opponent, but Modified Two Corners succeeded fairly well. In Chenier and Vanderford’s calculations, Water Works was the most desirable property, followed by Electric Co. and three railroads — B&O, Reading, and Pennsylvania. Mediterranean Ave. was last. Of the property groups, orange was most valuable, dark purple least. And going first yields a significant advantage.

“In order for everyone here to become Monopoly Moguls, we offer the following suggestions: If your opponent offers you the chance to go first, take it. Buy around the board in a defensive manner (that is at least one property per group). When trading begins, trade for the Orange-Red corner as well as for the Lt. Blue properties. They are landed on most frequently and offer the best return. The railroads and utilities offer a good chance for the buyer to raise some cash with which he may later develop other properties. Finally, whenever your opponent has a hotel on Boardwalk, never, we repeat, never land on it.”

(Thomas Chenier and Cathy Vanderford, “An Analysis of Monopoly,” Pi Mu Epsilon Journal 7:9 [Fall 1983], 586-9.)

August 27, 2016 | Entertainment · Science & Math

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.)

August 26, 2016August 25, 2016 | 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.)

August 25, 2016 | Language · Science & Math

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.”

August 24, 2016August 24, 2016 | Language · Science & Math

The Simson Line

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.

August 22, 2016September 14, 2016 | Science & Math

The Power of Prayer

In 1872 Francis Galton reflected that congregations throughout Britain pray every Sunday for the health of the British royal family. If prayer has tangible effects, he wondered, shouldn’t all this concentrated well-wishing result in greater health for its objects? He compared the longevity of royalty to clergy, lawyers, doctors, aristocracy and gentry, as well as other professions, and found that

[t]he sovereigns are literally the shortest lived of all who have the advantage of affluence. The prayer has therefore no efficacy, unless the very questionable hypothesis be raised, that the conditions of royal life may naturally be yet more fatal, and that their influence is partly, though incompletely, neutralized by the effects of public prayers.

He noted also that missionaries are not vouchsafed a long life, despite their pious purpose; that banks that open their proceedings with prayers don’t seem to receive any benefit from doing so; and that insurance companies don’t offer annuities at lower rates to the devout than to the profane. Certainly men may profess to commune in their hearts with God, he wrote, but “it is equally certain that similar benefits are not excluded from those who on conscientious grounds are sceptical as to the reality of a power of communion.”

(Francis Galton, “Statistical Inquiries Into the Efficacy of Prayer,” Fortnightly Review 12 [1872], 125-35.)

August 20, 2016August 19, 2016 | Religion · Science & Math

Head and Heart

In 2001 UC-San Diego sociologist David Phillips and his colleagues noted that deaths by heart disease seem to occur with unusual frequency among Chinese and Japanese patients on the 4th of the month. A study of death records revealed a 7 percent increase in cardiac deaths on that date, compared with the daily average for the rest of the week. And deaths from chronic heart disease were 13 percent higher.

One explanation is that the number 4 sounds like the word for “death” in Mandarin, Cantonese and Japanese, which causes discomfort and apprehension among some people. The effect is so strong that some Chinese and Japanese hospitals refrain from assigning the number 4 to floors or rooms. The psychological stress brought on by that date, the researchers suggest, may underlie the higher mortality.

They dubbed this the Baskerville effect, after the Arthur Conan Doyle novel in which a seemingly diabolical dog chases a man, who flees and suffers a fatal heart attack. “This Baskerville effect seems to exist in fact as well as in fiction,” they wrote in the British Medical Journal (PDF).

“Our findings are consistent with the scientific literature and with a famous, non-scientific story. The Baskerville effect exists both in fact and in fiction and suggests that Conan Doyle was not only a great writer but a remarkably intuitive physician as well.”

August 18, 2016August 17, 2016 | Literature · Science & Math

Desargues’ Theorem

If two triangles ABC and abc are oriented so that lines Aa, Bb, and Cc meet at a point, then the pairs of corresponding sides (AB and ab; BC and bc; and AC and ac) will meet in three collinear points.

The converse is also true: If the pairs of corresponding sides intersect in three collinear points, then the lines joining corresponding vertices will meet in a point.

August 13, 2016August 12, 2016 | Science & Math

Cultural Outreach

Scotland’s 1904 antarctic expedition made a unique contribution to science:

A number of emperor penguins, which were here very numerous, were captured. … To test the effect of music on them, Piper Kerr played to one on his pipes, — we had no Orpheus to warble sweetly on a lute, — but neither rousing marches, lively reels, nor melancholy laments seemed to have any effect on these lethargic phlegmatic birds; there was no excitement, no sign of appreciation or disapproval, only sleepy indifference.

— Rudmose Brown et al., The Voyage of the “Scotia,” 1906

As Ernest Shackleton was approaching Antarctica on December 18, 1914, “During the afternoon three adelie penguins approached the ship across the floe while Hussey was discoursing sweet music on the banjo. The solemn-looking little birds appeared to appreciate ‘It’s a Long Way to Tipperary,’ but they fled in horror when Hussey treated them to a little of the music that comes from Scotland.”

August 12, 2016November 4, 2024 | Oddities · Science & Math

Organic Chemistry

findig benzene

In a joke issue of the Berichte der Deutschen Chemischen Gesellschaft in 1886, F.W. Findig offered an article on the constitution of benzene in which he finds that “zoology is capable of rendering the greatest service in clearing up the behavior of the carbon atom”:

Just as the carbon atom has 4 affinities, so the members of the family of four-handed animals possess four hands, with which they seize other objects and cling to them. If we now think of a group of six members of this family, e.g. Macacus cynocephalus, forming a ring by offering each other alternately one and two hands, we reach a complete analogy with Kekulé’s benzene-hexagon (Fig. 1).

Now, however, the aforesaid Macacus cynocephalus, besides its own four hands, possesses also a fifth gripping organ in the shape of a caudal appendix. By taking this into account, it becomes possible to link the 6 individuals of the ring together in another manner. In this way, one arrives at the following representation: (Fig. 2).

“It appears to me highly probable that a complete analogy exists between Macacus cynocephalus and the carbon atom,” Findig wrote. “In this case, each C-atom also possesses a caudal appendix, which, however, cannot be included among the normal affinities, although it takes part in the linking. Immediately this appendix, which I call the ‘caudal residual affinity’, comes into play, a second form of Kekulé’s hexagon is produced; this, being obviously different from the first, must behave differently.”

(From John Read, Humour and Humanism in Chemistry, 1947.)

August 11, 2016August 11, 2016 | Humor · Science & Math