November 29, 2009

Running in the Rain

Many years ago, when I was a high school physics teacher, one of my favorite books was The Flying Circus of Physics by Jearl Walker. The book offered dozens of provocative questions about the everyday world to get you thinking about physics, whether you were cooking, flying, lazing at the beach, or getting caught in the rain.

This raises the question, "Should you run or walk when you cross the street in the rain without an umbrella?"

Walker noted that running would mean less time spent in the rain. However, running into some rain might make you wetter than if you had walked. He suggested doing a rough calculation, making the approximation that your body is a rectangular object. "Using such a model, can you tell if your answer (whether to run or walk) depends on whether the rain is falling vertically or at a slant?" Walker wrote.

The original edition of The Flying Circus of Physics included references but didn't provide any answers, so teachers and students were on their own. A later edition added answers, but often with the caution that the underlying physics hadn't yet been worked out satisfactorily.

Walker's solution to the "run or walk" question was to run as fast as possible if the rain were toward your front or overhead. If the rain were toward your back, you should run with a speed equal to the horizontal velocity of the rain (that is, along with the rain).

Many others have posed the same question and come up with roughly the same answer, as summarized in the following limerick by Matthew Wright from a 1995 New Scientist article.

When caught in the rain without mac,
Walk as fast as the wind at your back,
But when the wind's in your face
The optimal pace
Is fast as your legs will make track.

Some people have noted, however, that for a sufficiently weak tail-wind, it is actually better to run as quickly as possible rather than "as fast as the wind."

Moreover, shape matters, as Dan Kalman of American University (writing as Dank Hailman of Jamaicarain University) and Bruce Torrence of Randolph-Macon College (writing as Bruce Torrents of Raindrop-Macon College) point out in the article "Keeping Dry: The Mathematics of Running in the Rain," published in the October Mathematics Magazine.

Most previous analyses modeled the damp pedestrian as a rectangular solid. Kalman and Torrence considered an ellipsoidal traveler. Their results indicate that, under nearly all conditions, it is beneficial to run at top speed.

"Our recommendation, therefore, is to RUN in the rain unless you find yourself traveling in the perfect storm—where the tail-wind is half your top running speed, the cross-wind is minimal, and the rainfall is light," Kalman and Torrence conclude. "In such conditions, given the rounded features of the human body, it might make sense to dampen your pace (so to speak) from a run down to a speed that is just a bit faster than that of the tail-wind."

The authors offer the following advice:

When you find yourself caught in the rain,
while walking exposed on a plane,
for greatest protection
move in the direction
revealed by a fair weather vane.
Moving swiftly as the wind we'll concede,
for a box shape is just the right speed.
But a soul who's more rounded
will end up less drownded
if the wind's pace he aims to exceed.


Bailey, H. 2002. On running in the rain. College Mathematics Journal 33:88-92.

Deakin, M.A.B. 1972. Walking in the rain. Mathematics Magazine 45:246-253.

Hailman, D., and B. Torrents. 2009. Keeping dry: The mathematics of running in the rain. Mathematics Magazine 82(October):266-277.

Schwartz, B.L., and M.A.B. Deakin. Walking in the rain, reconsidered. Mathematics Magazine 46:272-276.

Walker, J. 1977. The Flying Circus of Physics (with Answers). Wiley.

November 8, 2009

Random Bias

When you're making an estimate, extraneous factors or irrelevant information can strongly bias your judgment, especially when the situation involves a lot of uncertainty.

Physicist and writer Leonard Mlodinow provided a vivid illustration of such bias during his recent colloquium talk on randomness, which he presented at the National Institute of Standards and Technology in Gaithersburg, Md. Mlodinow is currently a professor at Caltech. He is the author of The Drunkard’s Walk: How Randomness Rules Our Lives.

Your powers of estimation are easily influenced by minor things, random things that happen around you, Mlodinow argued.

To demonstrate, Mlodinow divided his audience into two groups. Each member of each group independently wrote down on a slip of paper a numerical answer to the question: How many countries are there in Africa? The slips were collected and the results tallied.

Answers from the first group averaged about 47; answers from the second group, about 27. Why the difference?

Each group also answered an introductory yes-or-no question that the other group did not see. The first group was initially asked: Are there more than 180 countries in Africa? The second group was initially asked: Are there more than 5 countries in Africa?

Most members of the audience of scientists and engineers probably had no clear idea what the correct answer is. They had to make a guess and apparently were influenced by the number that they saw in the first question. The results would undoubtedly have been different with, say, an audience of Africa scholars, who would likely be much more familiar with the continent.

Africa actually has about 50 countries, depending on how you count disputed territories and whether you include offshore island nations.

At the NIST talk, I was in the first group, and I knew that Africa had a lot of countries. I also recalled that the United Nations had nearly 200 members, so I based my estimate on what fraction of the total would be in Africa. I came up with 70, which was too high and was probably influenced by my seeing 180 in the first question.

Mlodinow has tried the same experiment with a variety of audiences, always obtaining a striking difference between the two groups. When he presented the talk last year to an audience at Google, for example, the estimates averaged about 65 and 30. (At Microsoft, the estimates had been 50 and 24.)

Mlodinow called this effect "anchoring bias." When making estimates, "be careful before you trust them," he warned.

One practical lesson, however, is that in negotiations it pays to ask high, whether you're fishing for a higher allowance or suing someone for damages. In the face of uncertainty, this posited sum may very well influence the size of the final outcome.

We take in, filter, and interpret a lot of data every day, Mlodinow noted. When the situation involves uncertainty or randomness, we often make mistakes or draw improper conclusions.

November 3, 2009

Cosmic Reflection

The universe began with a bang 13.7 billion years ago, so it was no surprise that a terse bang accompanied a video illustration of the event. This sonic boom was itself just one fleeing moment in a narrated, 40-minute symphony, titled Cosmic Reflection, which debuted on Nov. 2, 2009, at the Kennedy Center in Washington, D.C.

Composed by Nolan Gasser, the symphony was his tribute to the Fermi Gamma-Ray Space Telescope in its first year in orbit. In words, pictures, and music, the composition told the story of the universe, offering a striking lesson in modern astrophysics and cosmology, from the big bang to dark energy and cold death.

Written by Classical Archives founder Pierre R. Schwob and physicist Lawrence M. Krauss, the narration presented the known facts and theoretical speculations simply and clearly—offering a model of accessible exposition. The poetry was in the music, played crisply and impressively by the Boston University Symphony Orchestra, and in the video images provided by the NASA Goddard Space Flight Center: twisting tendrils and flickering rhythms; swinging galaxies and swirling tones; flashing stars and melodic fragments; cataclysmic collapses and percussive effects.

The large audience, blending the sensibilities of the arts, science, and engineering, gave the performance a standing ovation.

While experiencing the event, I couldn't help wondering what a narrated, illustrated symphony devoted to a mathematical theme would be like. Of course, astronomy itself has been a significant driving force in the development of mathematics. Isaac Newton, for one, wrote his immortal Principia Mathematica in answer to the question of what sorts of orbits would occur under an inverse-square force law. I wrote about these connections in my book Newton’s Clock: Chaos in the Solar System. And mathematics has been an invaluable tool for astrophysicists and cosmologists peering into the deepest reaches of the universe. So, mathematics could lay a claim to Cosmic Reflection, too.

What about the story of the prime numbers and the 150-year-old Riemann Hypothesis, a mathematical question for which the Clay Mathematics Institute has offered a $1 million prize for its solution? But that story isn’t complete; there is no proof yet, and no certainty that anyone will ever find a proof. And what images would you use?

Despite the pervasiveness of mathematical thought and its crucial role in underpinning scientific research, I am at a loss. What story, told in words, images, and music, could match the incredible epic that astronomers and physicists have forged from their observations over many millennia of spots of light in the sky?

August 31, 2009

Drilling a Square Hole

A mechanism for drilling square holes has to turn circular motion into square motion.

In one early attempt to create such a device, James Watts had the idea of rotating a Reuleaux triangle within a square. A Reuleaux triangle, named after mechanical engineer Franz Reuleaux (1829–1905), has the same width all the way around. Its shape is made from arcs of circles centered at the vertices of an equilateral triangle.

To make a Reuleaux triangle, draw three arcs of circles, with each arc having as its center one of an equilateral triangle's vertices and as its endpoints the other two vertices.

Like a circle, this rounded triangle fits snugly inside a square having sides equal to the curve's width no matter which way the triangle is turned. As it rotates, the curved figure traces a path that eventually covers nearly every part of the square. Watts started a company, Watts Brothers Tool Works in Wilmerding, Pa., to make square-hole drills based on this idea. The company is still in operation today.

However, the resulting drilled shape is not a perfect square. Its corners are slightly rounded.

Rotated inside a square, a Reuleaux triangle traces a curve that is almost a square.

Barry Cox (University of Wollongong) and Stan Wagon (Macalester College) have recently explored geometric solutions to the problem of drilling exact square holes. They describe their investigations in the article "Mechanical Circle-Squaring," published in the September College Mathematics Journal.

Cox and Wagon begin with a mechanical device originally presented in an anonymous 1939 article in the magazine Mechanical World. John Bryant and Chris Sangwin (University of Birmingham) revisited the design in their book How Round Is Your Circle? Where Engineering and Mathematics Meet (Princeton University Press, 2008) and built a physical model of the drill.

The geometric key is to use a variant of the classic Reuleaux triangle in which one vertex is rounded off. The starting point is an isosceles right triangle. In the completed construction, the vertex at the right angle traces out a small square when the entire figure rotates within a larger square.

In this variant of the classic Reuleaux triangle, the vertex C traces out the inner square (dashed lines) when the rotor, a curve of constant width, rotates so that it always lies within the outer square.

"If one places a cutting tool at point C . . . and turns the rotor so that it stays inside the large square, then C traces out an exact square and the cutting tool stays inside the larger square," Cox and Wagon write. "Thus the device can be viewed as a drill that drills an exact square hole, though we need to bring the construction into the third dimension to get a working model."

Using computer algebra tools, Cox and Wagon generalized the approach to obtain a roller for drilling a perfect hexagonal hole. In this case, the required shape is made up of six circular arcs.

A roller for drilling a perfect hexagonal hole is made up of six circular arcs, centered at O, A, X, O, Y, and F.

"Similar ideas work for the octagon, and it seems likely that they will extend to regular n-gons when n is even," Cox and Wagon conclude. "So the main unresolved problem is whether one can construct a device along these lines that will make a 3- or 5-sided hole."


Cox, B., and S. Wagon. 2009. Mechanical circle-squaring. College Mathematics Journal 40(September):238-247.

Peterson, I. 2003. Rolling with Reuleaux. MAA Online (Sept. 22).

August 20, 2009

The Geometric Spectacle of Water Fountains

Whether spraying graceful arcs of water into the air or letting water tumble down steep slopes, outdoor fountains draw attention. They can startle the eye and soothe the ear. They can offer cool respite from a day's travails.

Andrew J. Simoson of King College has pondered the factors that contribute to the visual impact of water fountains. What makes some fountains more spectacular than others? To Simoson, one ideal is the mathematical answer to the question: "For a given initial speed of water from a spigot or jet, what angle of the jet maximizes the visual impact of the water spray in the fountain?"

Simoson presents his analysis in the article "Maximizing the Spectacle of Water Fountains," published in the September College Mathematics Journal.

Jets of water spray out from the center of a fountain at Chevy Chase Circle in Washington, D.C. Photos by I. Peterson.

In many fountains, water sprayed from a jet or spewed from a spigot tends to follow a parabolic path. The particular arc is determined by the jet's angle and the speed at which water leaves the jet.

Simoson visualized the surfaces suggested by two arrays of water jets: one in which the jets are in a row and another in which they form a ring. For a circular array, the suggested surface is a surface of revolution of any of the parabolic streams about an appropriate vertical axis.

Simoson proposed two criteria for visual impact. In each of the two types of arrays that he examined, he posited that the visual impact is greatest when the volume enclosed by the suggested surface is a maximum with respect to the angle of the jets or when the surface area of the suggested surface is a maximum with respect to the angle of the jets.

In his analysis, Simoson assumed that the water streams are smooth parabolic arcs. He also ignored any friction effects. In the linear case, the problem simplifies to finding the angle that gives the maximum area and the maximum arc length for a single parabolic stream.

Jets spray water along paths that approximate parabolas.

Simoson's analysis shows that, for a linear array, the most spectacular fountains with respect to enclosed volume occur when the jets are inclined at 60°. However, with respect to surface area, the critical angle to achieve the most spectacular effect can be as low as 52.8°, depending on the difference in height between the start and end of an arc. For streams in which the end point is at half the height to which the water rises, the angle is about 53.5°.

For circular arrays of jets, the critical angles are less than 60° for both maximum volume and maximum surface area and differ from the results for linear arrays, going as low as 49.05°.

"Intuition may have suggested that the angles for the volume-spectacular fountain and surface-spectacular fountain might be the same as the angles that give a maximum area and a maximum arc length for solitary parabolic stream," Simoson notes. That doesn't happen.

"The reason must lie in the difference between a linear array of jets versus a circular array of jets," he explains. "Evidently, the advantage of having jets set at 60° in as far as accruing area under the streams and hence volume under the surface of revolution is eroded by the fact that adjacent streams get closer together as they near the fountain center, hence contributing less volume than intuition may have at first expected."

Simoson's analysis opens up the possibility of deliberately designing fountains that meet his criteria for spectacle—or checking existing fountains to see how close they come to the relevant criteria.

Portland, Ore., where MathFest 2009 took place earlier this month, has a variety of outdoor water fountains. Unfortunately, my photos of several of these fountains were not precise enough for me to check how closely they matched Simoson's criteria.

Water streams in parabolic arcs from a tubular fountain in Portland, Ore.

At the same time, fountains don't have to feature water jets in any sort of array to be spectacular or even aesthetically pleasing.

Water cascades down stone blocks in this striking Portland fountain.

And water jets can be combined with other mathematics—the Fibonacci sequence, for instance—to create intriguing effects, as seen in Helaman Ferguson's Fibonacci Fountain in Bowie, Md.

Helaman Ferguson's Fibonacci Fountain in Bowie, Md. Photo courtesy of Helaman Ferguson.


Simoson, A.J. 2009. Maximizing the spectacle of water fountains. College Mathematics Journal 40(September):263-274.

July 29, 2009

High School Musical and Formulas for Pi

Toward the end of the first paper that he published in England, famed Indian mathematician Srinivasa Ramanujan (1887–1920) offers three series representations of the inverse of the constant pi (1/π).

Amazingly, two of these formulas appear on a blackboard in a scene in the insanely popular Disney movie High School Musical, except that one of the formulas isn't quite right. In the movie, young star Vanessa Anne Hudgens plays brainy Gabriella Montez, who asks her teacher, "Shouldn't the second equation read sixteen over pi." The expression on the board reads 8/π.

"Sixteen over pi," the teacher replies. "That's impossible." She then whips out a calculator and somehow figures out that Gabriella is right. "I stand corrected," she concedes.

The correct version of Ramanujan's series for 1/pi that appears in High School Musical.

That bit of dialog near the beginning of the movie helps establish where Gabriella stands as a student at her high school. This movie moment also now figures in a paper published in the August-September issue of the American Mathematical Monthly. Nayandeep Deka Baruah, Bruce C. Berndt, and Heng Huat Chan provide a survey of Ramanujan’s series for 1/π and start off with the formulas that play a part in High School Musical.

An acknowledgement from the authors notes, "We are pleased to thank Si Min Chan and Si Ya Chan for watching High School Musical, thereby making their father aware of Walt Disney Productions' interest in Ramanujan's formulas for 1/π."

The paper originally appeared in a special issue of the Indian journal The Mathematics Student, published in 2007 by the Indian Mathematical Society in its centennial year.


Baruah, N.D., B.C. Berndt, and H.H. Chan. 2009. Ramanujan's series for 1/π: A survey. American Mathematical Monthly 116(August-September):567-587.

July 10, 2009

St. Louis Parabolas

Arches are commonplace architectural features; arches in the shape of parabolas are rather rare. One of the most striking instances of the use of parabolas in architecture can be found at the Priory Chapel of Saint Louis Abbey in Creve Coeur, Missouri.

Designed by Gyo Obata, the chapel’s circular façade consists of three tiers of concrete parabolic arches. The arches stretch upward from a grassy base; the top row forms a bell tower.
Photo by Jane Barnard.

Based in St. Louis, Obata was also responsible for the striking hyperboloid design of the James S. McDonnell Planetarium at the St. Louis Science Center. In addition, he designed the National Air and Space Museum in Washington, D.C., a structure that consists of four marble-encased cubes connected by three steel-and-glass atria.

A view of the National Air and Space Museum. Photo by I. Peterson.

The famous Gateway Arch, part of the Jefferson National Expansion Memorial in St. Louis, is not a parabola. Designed by Eero Saarinen, the structure is essentially an inverted catenary, described by the hyperbolic cosine function. Obata studied under Saarinen at the Cranbrook Academy of Art in Michigan.

June 18, 2009

From Hexagon to Square

The hexagonal terra cotta tiles that cover the floors of Metrorail stations are among the more distinctive features of the subway system serving Washington, D.C., and its neighbors.

Several years ago, the board in charge of Metro debated whether to replace the original hexagonal tiles with square ones. The terra cotta tiles had proved treacherous when wet, and manufacturers of safer tiles apparently made squares but not hexagons. And slicing off corners to turn squares into hexagons would be costly and wasteful.

In the end, Metro decided to test precast concrete paver tiles as a replacement for the original terra cotta tiles. Now, the verdict is in. The pavers proved more durable and did not become as slick when wet. Metro will use the new tiles in the construction of all future Metrorail stations and install them at other stations as part of its ongoing platform maintenance program.

The pavers are said to be square, but to help preserve the original look, they are imprinted with a hexagonal pattern and have a vaguely terra cotta color.

The new tiling unit is shown above. Its inscribed pattern consists of eight complete hexagons and parts of several others, for the equivalent of 14 hexagons. But there are other ways to divide a hexagonal tiling into "square" units. Is the chosen tile pattern the best one to create the desired repeating arrangement?

A side-by-side comparison of the original (right) and new tiles. WMATA

Metro describes the new pavers as being 2 feet by 2 feet. But can the pavers really be square, or is Metro (or the paver manufacturer) cheating a tiny bit to make it possible to create a hexagonal tiling out of "squares"? If the hexagons are truly to be regular, I get a slight difference between the height and width of a paver.

March 26, 2009

The Ladies' Diary

Odds bobs, ladies, what am I?
I'm at a distance, yet am nigh;
I'm high and low, round, short, and long,
I'm very weak, and very strong.
Sometimes gentle, sometimes raging.
Now disgusting, now engaging.
I'm sometimes ugly, sometimes handsome. . . .
I'm very dirty, very clean,
I'm very fat, and very lean,
I'm very thick, and very thin,
Can lift a stone, tho' not a pin. . . .

This riddle, written entirely in verse, goes on for about 30 lines. Submitted by a reader, it starts off the riddle section of the 1808 edition of The Ladies' Diary: or, Woman's Almanack. The Ladies' Diary was published annually, starting in 1704. Printed in London, it featured the usual stuff of almanacs: calendar material, phases of the moon, sunrise and sunset times, important dates (eclipses, holidays, school terms, etc.), and a chronology of remarkable events.

The little book's subtitle provides a glimpse of its purpose: "Containing New Improvements in ARTS and SCIENCES, and many entertaining PARTICULARS: Designed for the USE AND DIVERSION OF THE FAIR SEX."

Among those diversions were sections devoted to riddles (called enigmas), rebuses, charades, scientific queries, and mathematical questions. A typical volume in the series included answers submitted by readers to problems posed the previous year and a set of new problems, nearly all proposed by readers. Both the puzzle and the answer (revealed the following year) were often in verse.

In a paper just published in Historia Mathematica, Joe Albree and Scott H. Brown of Auburn University review the mathematical contribution of The Ladies' Diary, particularly the leading role it played in the early development of British mathematical periodicals.

"From the first decade of the 18th century and for many decades following, The Diary was a pioneer in more than one respect," Albree and Brown write. "With unfailing regularity, starting in 1708, it presented an array of mathematical problems and their solutions to a wide range of readers."

"From the very start women were encouraged to participate fully in The Diary's mathematical program," they add.

The Ladies' Diary was founded by John Tipper (before 1680–1713), in part to advertise the Bablake School in Coventry, England, where he was master, and to help support his family financially. The publication also represented his recognition of a need for an almanac that catered to women.

"Tipper's attitude toward women was often remarkably progressive," Albree and Brown remark. The cover of each issue featured a picture of a prominent English woman, beginning with Queen Anne in 1704.

The publication was also a success. In its second year, about 4,000 copies of The Diary were sold; circulation reached about 7,000 in 1718 and, in the middle of the 18th century, amounted to around 30,000 copies a year.

The readership was diverse, and the publication attracted a wide range of contributors of both problems and solutions. However, because many contributor used pseudonyms, it's difficult to determine how many were female.

"The earliest mathematical questions in The Diary were puzzles, enigmas solved by numbers, and such contrived problems continued to appear through the whole life of The Diary," Albree and Brown say. "However, very quickly, the level of difficulty and the seriousness of many of the questions in The Diary increased, and The Diary became one of the participants in the popularization of mathematics and the Newtonian sciences in the 18th century."

Note that The Ladies' Diary had been founded while Isaac Newton (1643–1727) was still alive.

By the 19th century, some of the proposed problems were quite sophisticated and occasionally called for knowledge of contemporary issues and advances in mathematics or physics. The Diary even attracted some foreign contributions. Here's a geometry question submitted in 1830 by French mathematician C.J. Brianchon (1783–1864).

The six edges of any irregular tetrahedron are opposed two by two, and the nearest distance of the two opposite sides is called breadth; so that the tetrahedron has three breadths and four heights. It is required to demonstrate that, in every tetrahedron, the sum of the reciprocals of the squares of the breadths is equal to the sum of the reciprocals of the squares of the heights.

Some questions were open-ended. Here's one from 1822:

Required a better method than has yet been published of finding x in the equation xx = a.

I came across the 1808 and 1809 editions of The Ladies' Diary among the volumes in a remarkable collection of books at the library of the University of Calgary in Alberta. The Eugène Strens Recreational Mathematics Collection contains more than 6,000 items, including books, periodicals, newspaper clippings, and manuscripts devoted to recreational mathematics (in a very broad sense) and its history.

"The bulk of mathematics has really always been recreational," says mathematician Richard K. Guy, who was instrumental in bringing to the University of Calgary material collected by the late Eugène Strens, an engineer, amateur mathematician, and friend of the artist M.C. Escher. "Only a tiny fraction of all mathematics is actually applied or used."

Here's a riddle from the 1809 edition of The Ladies' Diary:

I'm a singular creature, pray tell me my name,
I partake of an Englishman's freedom and fame;
I daily am old, and I daily am new,
I am praised, I am blam'd, I am false, I am true;
I'm the talk of the nation, while still in my prime,
But forgotten when once I've outlasted my time.
In the morning no Miss is more coveted than I,
In the evening no toy thrown more carelessly by.
Take warning, ye fair! I, like you, have my day,
And, alas! You, like me, must grow old and decay.

A typical rebus provided clues for a starting word, which was then modified step by step to produce the final word:

My whole's a small but luscious fruit;
Take off my head and then you'll see
What sinful men sometimes commit,
That brings them to the fateful tree.
Two letters now you may transpose,
But place them both with care,
Another luscious fruit will then,
Most plainly soon appear.


Take two-fifths of what, seen on Delia's fair face,
Always adds to her charms an ineffable grace;
These selected with care, and with art combined,
Will produce a Diarian of genius refin'd.

Scientific queries covered a wide range of subjects:

Query: A drop of oil let fall on a wasp, kills it in less than a minute; query: how is this effected?

Query: Thirteen years have elapsed since the Northern Lights have made their appearance. How is their absence accounted for?

The mathematical questions tended to focus on geometry, and they were reminiscent of the types of word problems found in math textbooks of a few generations ago.

A circular vessel, whose top and bottom diameter are 70 and 92, and perpendicular depth 60 inches, is so elevated on side that the other becomes perpendicular to the horizon; required what quantity of liquor, ale measure, will just cover the bottom when in that position.

The book didn't provide a diagram.

The math questions featured in The Ladies' Diary were of sufficient interest that collections of them were published in four volumes covering the years 1704 to 1817.

I wonder if Jane Austen (1775–1817) or Ada Lovelace (1815–1852) ever pondered the puzzles posed by The Ladies' Diary?

Riddles: water; newspaper.
Rebuses: grape, rape, pear; smile, smart.
Math question: Very nearly 189 gallons.


Albree, J., and S.H. Brown. 2009. "A valuable monument of mathematical genius": The Ladies' Diary (1704-1840). Historia Mathematica 36(February):10-47.

Almkvist, G., and B. Berndt. 1988. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, pi, and The Ladies' Diary. American Mathematical Monthly 95(August-September):585-608.

Guy, R.K., and R.E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Mathematical Association of America.

March 12, 2009

Rock-Paper-Scissors for Winners

The simple game of rock-paper-scissors has a long history as a way of settling disputes, whether in the playground or at the conference table.

In the playground version of rock-paper-scissors, each of two players makes a fist. On a count of three, each player simultaneously puts down a fist, which means rock, a flat hand, which means paper, or two fingers in a "V," which means scissors. The following non-transitive rules decide the winner: Rock beats scissors, scissors cut paper, and paper wraps rock. If both players make the same gesture, the bout ends in a tie.

Is there a winning strategy for this game? It doesn't make sense to show the same configuration each time. An alert opponent would quickly learn to anticipate your move, make the appropriate response, and always win. A similar danger lies in following any other sort of pattern. Thus, unless you can find a flaw in your opponent's play, your best bet is to mix the three choices randomly.

But without the help of a randomizer of some sort, people aren't very good at making random choices on their own. In the article "Winning at Rock-Paper-Scissors" in the March 2009 College Mathematics Journal, Derek Eyler, Zachary Shalla, Andrew Doumaux, and Tim McDevitt present strategies for defeating people who are poor generators of random sequences.

Suppose, for example, that a player tends to choose one option (symbol) more often than the others. According to McDevitt and his coauthors, winning play generalizes to the following strategy:

* Never choose the symbol that loses to the most likely symbol.
* Choose the most likely symbol if the symbol that it beats has probability greater than 1/3.
* Otherwise, choose the symbol that beats the most likely symbol.

To see what happens in practice, McDevitt and his associates collected data from 119 people who played 50 games of rock-paper-scissors against a computer using this optimal strategy. Their subjects had a strong preference for rock: 55.5% started with rock, 32.8% with paper, and 11.8% with scissors.

"Symbol choices beyond the first seem to depend on previous choices," the authors note. "Players seem to have a distinct preference for repeating plays."

For example, a player who chose paper on one trial had a .421 probability of repeating this play on the next trial, well above 1/3.

So, when playing against such a player, your best bet is a strategy of always playing the symbol that defeats the symbol that your opponent previously played. But that only works in the short term. In the long run, you would end up choosing the same symbol every time, and your opponent might catch on.

"A more sophisticated approach analyzes an opponent's previous two choices," McDevitt and his colleagues write. Experimental results suggest that people have a tendency to repeat symbols or to cycle through the symbols.

The researchers developed a computer program that bases its choice of symbol adaptively on its human opponent's previous two choices, updating the probabilities of different pairs of choices every time its human opponent chooses a symbol.

In one test that involved 241 participants each playing 100 games against the computer, the computer won 42.1%, lost 27.7%, and tied 30.2% of individual games. That's significantly better than playing randomly would achieve.

"In closing, we note that deviating from the optimal strategy is always dangerous because a superior strategy always exists," the researchers warn.

In the test, for example, two players defeated the program 100 times out of 100. "Conversations with those two players revealed that one of them stopped and started the match repeatedly until he found the perfect strategy through trial and error," McDevitt and his colleagues note, "and the second accessed and analyzed the program to determine the computer's strategy."

You can try beating a computer implementation of the strategy described above at McDevitt's "Rock-Paper-Scissors" Web page.


Beasley, J. 1990. The Mathematics of Games. Oxford University Press.

Eyler, D., Z. Shalla, A. Doumaux, and T. McDevitt. 2009. Winning at rock-paper-scissors. College Mathematics Journal 40(March):125-128.

Peterson, I. 2002. Mating games and lizards. In Mathematical Treks: From Surreal Numbers to Magic Circles. Mathematical Association of America.

February 20, 2009

The Remarkable Miss Mullikin

The name Anna Margaret Mullikin (1893-1975) doesn't appear on the MAA's "Women of Mathematics" poster. Her biography is not among those posted on the Agnes Scott College pages featuring women mathematicians.

"She is virtually unknown today, but . . . we believe she deserves greater recognition," Thomas L. Bartlow of Villanova University and David E. Zitarelli of Temple University contend in the February American Mathematical Monthly. Their article, "Who Was Miss Mullikin?", presents a compelling case for lauding the accomplishments of this mathematician and high school mathematics teacher.

Mullikin was the third Ph.D. student of R.L. Moore (1882-1974), a prominent figure in 20th-century mathematics who founded his own school of topology and advocated a teaching style that encourages students to solve problems using their own skills. She was the first of Moore's students to write a dissertation on topology, a pioneering work that dealt with connected sets.

Mullikin's dissertation was her only published mathematical research, which appeared in 1922. Nonetheless, her results had a "catalytic effect" on the fledgling field of point-set topology, Bertlow and Zitarelli point out. Mullikin's work "inspired a decade of intense investigations leading to applications and generalizations by two of the leading schools of topology at that time."

Mullikin obtained her A.B. degree in 1915 from Goucher College in Baltimore. She taught mathematics for three years before beginning graduate study at the University of Pennsylvania, where she quickly came to Moore's attention. When Moore left in 1920 for the University of Texas, he arranged to have her appointed as an instructor so that she could complete her thesis under his guidance. After her instructorship ended, Mullikin returned to the University of Pennsylvania to complete requirements for her degree.

Mullikin's dissertation, "Certain theorems relating to plane connected point sets," appeared in the September 1922 Transactions of the American Mathematical Society.

After completing her degree, Mullikin worked for the Philadelphia school district as a high school teacher, ending up at Germantown High School, where she taught until she retired in 1959. "During this 36-year tenure she earned a reputation as a demanding, sympathetic, and effective teacher of mathematics," Bartlow and Zitarelli note.

Mullikin "identified and encouraged students of strong mathematical ability, taught a meticulous and orderly approach to mathematics to all hers students, and tailored her lessons to the abilities of individual students," the authors conclude. "Although her pupils were unaware of her earlier exploits and some of them did not even know that she held a Ph.D., they benefited by experiencing firsthand a brilliant and serious mathematical mind at work."

Bartlow and Zitarelli provide many more details of Mullikin's life and career in their article. Belated recognition of Mullikin's mathematical work also comes in a new book, Pioneering Women in American Mathematics: The Pre-1940 Ph.D.s by Judy Green of Marymount University and Jeanne LaDuke of DePaul University, published by the American Mathematical Society.

Bartlow and Zitarelli mention three other women who received their mathematics degrees in 1921-1922. Margaret Buchanan (1885-1965) graduated from West Virginia University, then did her dissertation at Bryn Mawr College under Anna Pell Wheeler (1883-1966). She returned to teach at West Virginia for the rest of her career. Claribell Kendall (1889-1965) graduated from the University of Colorado, did her dissertation under Ernest Wilczynski (1876-1932) at the University of Chicago, then taught at Colorado for the remainder of her career. Eleanor Pairman (1896-1973) went to Radcliffe College, where she obtained her Ph.D. under George David Birkhoff (1884-1944). She married fellow graduate student Bancroft Huntington Brown in 1922, and Brown ended up at Dartmouth College. Pairman enjoyed teaching but had little opportunity to do so, trapped in a males-only college community.

Pairman "is the only one who married and the only one who published anything other than her own dissertation," Bartlow and Zitarelli remark. "The others, like Mullikin, remained single and embarked on teaching careers, albeit at their home universities."


Bartlow, T.L., and D.E. Zitarelli. 2009. Who was Miss Mullikin? American Mathematical Monthly 116(February):99-114. Preprint.

Green, J., and J. LaDuke. 2009. Pioneering Women in American Mathematics: The Pre-1940 Ph.D.s. American Mathematical Society. Supplementary material.

February 17, 2009

Trouble with Wild-Card Poker

Poker originated in the Louisiana territory around the year 1800. Ever since, this addictive card game has preoccupied generations of gamblers. It has also attracted the attention of mathematicians and statisticians.

The standard game and its many variants involve a curious mixture of luck and skill. Given a deck of 52 cards, you have 2,598,960 ways to select a subset of five cards. So, the probability of getting any one hand is 1 in 2,598,960.

A novice poker player quickly learns the relative value of various sets of five cards. At the top of the heap is the straight flush, which consists of any sequence of five cards of the same suit. There are 40 ways of getting such a hand, so the probability of being dealt a straight flush is 40/2,598,960, or .000015. The next most valuable type of hand is four of a kind.

The table below lists the number of possible ways that desirable hands can arise and their probability of occurrence.

The rules of poker specify that a straight flush beats four of a kind, which tops a full house, which bests a flush, and so on through a straight, three of kind, two pair, and one pair. Whatever your hand, you can still bet and bluff your way into winning the pot, but the ranking (and value) of the hands truly reflects the probabilities of obtaining various combinations by random selections of five cards from a deck.

Many people, however, play a livelier version of poker. They salt the deck with wild cards—deuces, jokers, one-eyed jacks, or whatnot. The presence of wild cards brings a new element into the game, allowing such a card to stand for any card of the player's choosing. It increases the chances of drawing more valuable hands.

Using wild cards also potentially alters the ranking of various sets of cards. You can even obtain five of a kind, which typically goes to the top of the rankings.

A while ago, mathematician John Emert and statistician Dale Umbach of Ball State University took a close look at wild-card poker. Wild cards can alter the game considerably, they wrote in a 1996 article in Chance describing their findings. "When wild cards are allowed, there is no ranking of the hands that can be formed for which more valuable hands occur less frequently," the authors argued.

In other words, when you play with wild cards, you can't rely on a ranking of hands in the order of the probability that they occur as you can when there are no wild cards. Magician and card expert John Scarne made a similar observation in his book Scarne on Cards, first published in 1949.

In a 1996 article in Mathematics Magazine, Steve Gadbois also concluded that wild cards mess up logical, probability-based poker play, producing all sorts of anomalies or paradoxes in the ranking of different hands. "The more one looks, the worse it gets," he remarked.

Wild cards increase the number of ways in which each type of hand can occur. For example, with deuces wild, four of a kind occurs more than twice as often as a full house. So, modifying the rules to rank a full house higher than four of a kind might produce a more consistent result.

A player, however, often has a choice of how to declare a hand and that means assembling the strongest possible combination allowed by the given rules. Thus, if a full house ranks higher than four of a kind, and a player has a wild card allowing him or her to choose either a full house or four of a kind, the full house will inevitably come up more often than four of a kind!

"There is no possible ranking of hands in wild-card poker that is based solely on frequency of occurrence," Emert and Umbach demonstrated. The researchers also examined alternative ranking schemes. They found that whatever the wild-card option, the standard ranking proves to have fewer inconsistencies than the alternatives.

Emert and Umbach then went on to see if there exists a better way of ranking the hands. They proposed a scheme that takes into account the fact that certain hands can be labeled in several ways. For example, any wild-card hand declared as a full house can also be considered as two pair, three of a kind, or even one pair or four of a kind.

The authors define a quantity called the inclusion frequency, which gives the number of five-card hands that can be declared as such for each type of hand. Rankings based on this number give hands with smaller inclusion frequencies a higher position in the list. In standard poker, this method leads to the traditional rankings. Wild-card variants show a slightly different order. Interestingly, one result of this new ranking criterion is that the greater the number of wild cards, the more valuable a flush becomes.

"We believe that the use of the 'inclusion' ranking of the hands presents a more consistent game than deferring to ordinary ranking," Emert and Umbach declared.

Of course, these analyses don't really take into account the complexity of what actually happens in a poker game. You're not likely to be computing probabilities as you play. It may be much more advantageous for you to put on your best poker face and bluff as much as you think you can get away with.

In discussion of simple games that involve bluffing, John Beasley, in The Mathematics of Games, wryly counsels: "Do not think that a reading of this chapter has equipped you to take the pants off your local poker school. Three assumptions have been made: that you can bluff without giving any indication, that nobody is cheating, and that the winner actually gets paid. You will not necessarily be well advised to make these assumptions in practice."

Some aspects of poker are beyond the reach of mathematics.


Beasley, John D. 1989. The Mathematics of Games. Oxford, England: Oxford University Press.

Emert, John, and Dale Umbach. 1996. Inconsistencies of "wild-card" poker. Chance 9(No. 3):17-22.

Gadbois, Steve. 1996. Poker with wild cards—A paradox? Mathematics Magazine 69(October):283-285.

Packel, Edward W. 1981. The Mathematics of Games and Gambling. Washington, D.C.: Mathematical Association of America.

Scarne, John. 1991. Scarne on Cards. New York: New American Library.

For information on poker odds, see "Poker Odds for Dummies" (

February 1, 2009

Pondering an Artist's Perplexing Tribute to the Pythagorean Theorem

The cover illustration of the January 2009 issue of The College Mathematics Journal (CMJ) has perplexed—even disturbed—a number of people. The cover features a photo of a 1972 work by prominent contemporary artist Mel Bochner titled Meditation on the Theorem of Pythagoras.

The artwork references the idea of relating the lengths of the sides of a 3-4-5 right triangle to the areas of the squares on those sides. To create the piece, Bochner used chalk and hazelnuts placed directly on the floor, materials that would have been familiar to Pythagoras. The construction represented his response to a visit to a temple in Metapontum, the city where Pythagoras purportedly died.

For some, the artwork represents a mathematical bungle. A contributor to the 360 blog, for example, wrote recently: "If the intent was to illustrate a right triangle, then the arrangement of hazelnuts is off." 360 is the unofficial blog of the Nazareth College math department in Rochester, N.Y., and features contributors from the college, local rival St. John Fisher College, and elsewhere.

"To my eye," the commenter continued, "the hazelnut grids look exactly like pins on a Geoboard, or lattice points in the plane. And given that perspective on this image, we see a 2-3-4 triangle, an obtuse triangle, and squares of area 4, 9, and 16." So, from that viewpoint, the mathematics isn't correct.

The commenter then presented his own version of what the artist might have meant in illustrating "the fundamental ideas of relating the lengths of the sides of a triangle to the areas of the squares on those sides using readily available materials."

In the end, the critic wrote, "I find Bochner's Meditation rather confusing, and to some extent disappointing."

In response, CMJ Editor Michael Henle of Oberlin College noted that Bochner's creation is still "marvelously evocative" of the Pythagorean Theorem, a common thread that links several articles in the January issue of CMJ. "This is, after all, a work of art, not mathematics," he added.

Moreover, the Pythagorean Theorem "is more than a piece of mathematics," Henle said. It is "also a piece of our culture and history over 2500 years old." The theorem, he noted, "still says what it says and Meditation on the Theorem of Pythagoras says something different."

Mel Bochner had his own rejoinder. He had visited the deserted temple on a cold and wet day in 1972, finding it little more than a few reconstructed columns, some ancient debris, and scattered building stones. Nonetheless, he strongly sensed the presence of Pythagoras and had the urge to commemorate that feeling.

Remembering his 10th-grade geometry (32 + 42 = 52, or 9 + 16 = 25), Bochner gathered 50 small stones from the temple debris and laid them down. But when he created his pattern, he found that he had three stones left over. Finally, it dawned upon him that the surplus came from counting the corners of the triangle twice.

"What I had stumbled upon was that physical entities (stones) are not equatable with conceptual entities (points)," Bochner said, "or the real does not map onto the ideal."

That's "why the title of the work is Meditation on the Theorem of Pythagoras and not simply Theorem of Pythagoras," Bochner noted, "and also why art is not an illustration of ideas but a reflection upon them."

Bochner welcomed the rediscovery of this "discrepancy" so many years after he had created the artwork. Yet he also wondered "about the unwillingness to assume that I already knew what they had just discovered (do mathematicians still think all artists are dumb?) and not take the next step and ask themselves if it might have been intended to be 'confusing.'"

Henle is pleased with the ongoing debate. "This is the kind of thing I hoped would happen," he said.

January 28, 2009


Suppose you want to pile identical bricks on the edge of a table so that the stack juts out as far as possible past the table's edge.

One solution is to pile the bricks one on top of the other, with each brick sticking out slightly farther than the one beneath it. Ideally, assuming vertical forces and no friction, the top brick juts out from the one below it by half its length, the second by a quarter, the third by a sixth, and so on. The nth brick from the top would overhang the one below it by 1/2n. The number of brick lengths by which the nth brick extends beyond the table edge is ½(1 + ½ + 1/3 + . . . 1/n)

Credit: Paterson and Zwick, American Mathematical Monthly

In principle, you can make the stack stick out as far as you want, but it could take a huge number of bricks to achieve the desired overhang.

Many people assume that this solution is optimal, but that’s true only if the bricks are stacked one on top of the other. What if more than one brick can lie atop another brick?

Mike Paterson of the University of Warwick and Uri Zwick of Tel Aviv University have now demonstrated that, when you allow such multiple stacking, you can get more overhang for the same number of bricks. They report their results in the article "Overhang," published in the January American Mathematical Monthly.

"Without this restriction, blocks can be used as counterweights, to balance other blocks," Paterson and Zwick write. "The problem then becomes vastly more interesting, and an exponentially larger overhang can be obtained."

The resulting optimal patterns for given numbers of bricks are fascinating—and somewhat unpredictable.

Optimal stacks for eight, nine, and ten bricks. The lightly shaded bricks in these stacks form the support set, while the darker bricks form the balancing set.

Paterson and Zwick used numerical methods to find the optimal stacks for up to 30 bricks. "While there seems to be a unique optimal placement of the blocks that belong to the support set of an optimal stack, there is usually a lot of freedom in the placementmof the balancing blocks," they remark.

Constructions of n bricks described as "parabolic," "vase," and "oil lamp" produce the best overhang, with each construction having an overhang of order n1/3.

A parabolic stack of 111 bricks produces an overhang of three bricks.

Paterson and Zwick, now joined by Yuval Peres of the University of California, Berkeley, Mikkel Thorup of AT&T Labs–Research, and Peter Winkler of Dartmouth College, describe additional findings on the stacking problem in their article "Maximum Overhang," to be published in the Monthly.

January 13, 2009

Statistical Wear

Marks on objects can provide intriguing statistical glimpses of usage patterns. The darkened leaves of a well-thumbed book may point to favorite passages; the distinctive hollows of oft-traversed steps suggest the characteristic tread of countless feet.

Last year, while I was at East Tennessee State University, I happened to notice a particularly striking example of such "statistical wear" on the door to the men's restroom, just down the hall from my office. Entry to the restroom was by a swinging door, which opened inward with a push.

Countless hands pushing on the door had worn away the brown stain in one particular area, reflecting where men had preferred to place a hand. The result was a roughly circular spot—a two-dimensional statistical distribution—with the most wear in the middle and progressively less wear away from the center.

Countless hands have worn away the stain in one particular spot on the swinging door to a men's restroom. Photos by I. Peterson.

Two factors probably contributed most to the two-dimensional pattern: the height of the individuals pushing on the door and perhaps some preference for how much force to apply (less force would be required to push open the door farther away from the hinges). Curiously, the pattern largely misses a brass plate that was likely supposed to be the target.

What pattern would you expect to see on a nearly identical swinging door to the women's restroom?

The pattern is similar, but it is lower and a little closer to the door's edge, reflecting a lower average height and a greater preference for pushing with less force. As a result, the pattern has a significantly greater overlap with the door's brass plate.

The door to the women's restroom also has a distinctive two-dimensional wear pattern.

Statistician Robert W. Jernigan of American University has been collecting such "visualizations" of statistical concepts for many years, from the pattern created on a brick wall by a leaking downspout to oil stains on a parking lot. His fascinating "Statpics" blog is devoted to images that illustrate statistical ideas.

Jernigan's paper, "A Photographic View of Cumulative Distribution Functions," appeared in the March 2008 Journal of Statistics Education.