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" (Cardschat.com).

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.