## Cutting a Pie is No Piece of Cake

Posted on Mar 30, 2007

Cutting pie is harder than slicing cake, at least if you want to do it fairly. Bizarre as that may sound, mathematicians have proven it.

Julius B. Barbanel of Union College and political scientist Steven J. Brams of New York University have long studied the classical mathematical problem of cake cutting. The question is this: Suppose you have an elaborately decorated sheet cake, with chocolate icing here, vanilla icing there, a cherry in one spot, and coconut sprinkled about. Some people may prefer particular portions of the cake. How do you divide the cake so that everyone gets a fair slice? The researchers are now asking the same question about pies.

"People started thinking of cake cutting and now pie cutting as interesting recreational issues," says Brams. "But in fact, things like divorce settlements and border disputes could be, if not settled, ameliorated by this way of thinking." Cake works as a metaphor for any good that can be divvied up.

Cutting a decorated sheet cake is the focus of complex, ongoing mathematics research. It might appear "simple as pie," however, compared with the challenges of cutting a decorated pie.

The difference lies in the cutting. We slice pies into wedges, but we cut sheet cakes into rectangles.

Mathematicians simplify cake cutting by assuming that all slices are perpendicular to one particular side of a rectangular cake, with no crosswise cuts. Following that rule, there is only one way to split a rectangular cake into two equal-size pieces.

For a pie, however, there are an infinite number of ways, because there an infinite number of lines going through the center of a circle. That simple difference leads to a world of mathematical trouble.

Furthermore, mathematical analysis of cutting decorated cakes and pies assumes that the portions are not necessarily of equal size. That complicates the challenges of cutting fair sizes of cake and makes pie cutting even more complex, if not impossible.

Barbanel and Brams wanted to find a way of cutting a pie that is "envy-free," meaning that each person is at least as happy with his own piece as he would be with anyone else's. They also wanted to make sure the division is "efficient," so that no other way of dividing the pie would be better for one person without becoming worse for others.

For cake, there are procedures for finding, or at least approximating, envy-free and efficient cuts for any number of people. But for pie, the situation is more difficult, the researchers found. Splitting a decorated pie between two people is not so tough, but creating fair shares for more than two people may be impossible.

For two people, the researchers found that it is possible to cut slices that are not only envy-free and efficient, but also "equitable." For example, you might get a larger piece than I get, but I may think that I got 60 percent of the value of the pie because I got the side with all the coconut, while you think you got 60 percent of the value of the pie because you got the side with the cherry on it. Since we each think we got 60 percent, it's "equitable."

Although researchers know that equitable division exists, they don't know how to produce it. They know how to cut a pie into two pieces in a way that is envy-free and efficient, but not necessarily equitable.

The method is like the classical "I cut, you choose" approach to cutting a cake: I cut the cake into two pieces that I perceive to be equal in value if not in size. Then, you choose the piece you prefer. Neither of us will envy the other, and there will be no way of increasing one person's share without decreasing the other's share. But it probably won't be equitable: From my perspective, I will get half the value of the cake. However, since you get to choose between the two pieces, you may perceive your piece as greater in value.

Applying the method to pies is more complicated, because there are infinitely many ways to cut a pie into two pieces that I perceive as equal in value. If the pie had a clock face and the clock's two hands were knives, pointing the hands to 12 and 6, for example, or to 10 and 4 would produce two equal-size pieces. But if there is a special piece of chocolate beside the 2 on the clock face, I might point the hands at, say, the 1 and the 3 if I thought a small piece of pie with the chocolate would have the same value as a far larger piece with no chocolate.

If you and I were using "I cut, you choose" to divide a pie, we could use a method that is like using knives as clock hands. First, I would position the end of one knife at the center of the pie and point the knife toward 12 o'clock. Then I would put the second knife in a position going from the center to another point on the edge, so as to produce two pieces I perceive as equal in value. Then, you would note your perceived value of each piece.

Next, I would rotate the first knife a bit, say, to one o'clock. I would again adjust the second knife to suggest two pieces that I think are of equal value. You would again note your perceived value of each piece. Once we've gone all the way around the pie, you'd tell me which positions of the knife gave you the piece with the most value, and I'd cut there.

In theory, we would need to do this an infinite number of times, because there are an infinite number of points on the edge of a pie. Mathematicians work with equations that represent an endless version of the "I cut, you choose" procedure.

As the player who does the "choosing," you will most likely get a slice you think is worth more than half the value of the pie. As one who does the "cutting," I will get a slice I think is worth exactly half. So the division isn't equitable, even though neither player "envies" the other and it's impossible to increase one player's share without decreasing the other's.

If at least four people are sharing a pie, the situation is much more challenging. The researchers show that sometimes it is impossible to cut the pie so that it's both envy-free and efficient, much less equitable.

Oddly, for three persons, no one knows whether it's even possible to have an envy-free, efficient division. "We worked and worked and worked at it," says Barabanel. So far, they haven't figured it out. "There are weird situations in math like that sometimes," he says, where the middle dimensions are the hardest to figure out. The Poincarï¿½ Conjecture is one example.

Will pie-cutting turn out to have the same kinds of real world applications that cake cutting has? "I think it's a bit of a stretch," Brams says. "But I would say that if you're trying to divide land on an island and you want people to have pieces of the shoreline, then pie cutting is better."

And, of course, it's a relief to learn that fair-minded mathematicians won't have to put up with rectangular pies anymore.

## Soledad O’Brien to discuss diversity

Posted on Mar 29, 2007

Soledad O'Brien, co-anchor of CNN's popular American Morning, will speak Monday, April 9 at 7 p.m. in the Nott Memorial as part of the Presidential Forum on Diversity. Her talk, “Diversity: On Television, Behind the Scenes and in our Lives,” is free and open to the public.

O'Brien joined CNN in July 2003 and has since covered major national and international stories. She was the only broadcast journalist permitted to travel with First Lady Laura Bush on her trip to Moscow in fall 2003. The following year, she was among a handful of CNN anchors sent to Puhket, Thailand, to cover the tsunami that claimed more than 155,000 lives.

Before joining CNN, O'Brien had been at NBC News since 1991, where she contributed reports for the Today Show and weekend editions of NBC Nightly News. She was anchor of Weekend Today since July 1999.

In 2003, she covered the Space Shuttle Columbia disaster, and she later anchored NBC's weekend coverage of the war in Iraq. In 1998, she covered Pope John Paul II's historic visit to Cuba.

A graduate of Harvard University, O'Brien began her career as an associate producer and news writer at the then-NBC affiliate, WBZ-TV in Boston. She also worked at MSNBC and at KRON in San Francisco.

She won a local Emmy for her work as a co-host on Discovery Channel's The Know Zone. She has been named to numerous “best” lists, including People magazine's “50 Most Beautiful People” (both English and Spanish versions); Crain's Business Reports' and Essence magazine's “40 Under 40”; and Irish American Magazine's “Top 100 Irish Americans.”

She is a member of the National Association of Black Journalists and the National Association of Hispanic Journalists. She also writes a bi-monthly column for USA Weekend magazine on parenting.

## Borowick photo a winner

Posted on Mar 29, 2007

Nancy Borowick '07 has received an honorable mention in a national student photography contest sponsored by Photographer's Forum magazine and Nikon. Her black-and-white image, “Reann” depicts a group of children she taught while on a term abroad in Barbados last year.

The 27th Annual College Photography Contest drew more than 28,000 entries from the United States, Canada and other countries worldwide. Borowick's photograph, among 100 honorable mentions, will be published in the Best of College Photography Annual 2007 in June. The publication is used in college libraries and in photography, art and graphic design courses.

There were 16 winners in first through fourth places. All of the winners and honorable mentions were chosen by judges Karen Sinsheimer of Santa Barbara Museum of Art, Steve LaVoie of the Art Center College of Design and Armando Flores of Nikon U.S.A.

Borowick, of Chappaqua, N.Y., is an organizing theme major who is combining courses in Anthropology, Visual Arts and Modern Languages. Her theme is titled “Representations of Culture and Identity.”

Last fall, she spoke about her work in Barbados at the 11th Annual Armand and Donald Feigenbaum Forum on “The Global Imperative: Approaches to Internationalizing the Union Experience.” Her digital artwork was featured recently at the Wikoff Student Gallery in the Nott Memorial as part of “Time Capsule 2006,” an array of works made by students in Visual Arts Professor Fernando Orellana's Intro to Digital Art class.

She has also participated in a number of opportunities for photography students at Union. In 2005, she took part in the Maine Photographic Workshops through the Beyond the Gates Fund, which supports students in their study of studio and visual arts outside the College. The fund is supported by a donation from an anonymous alumnus.

## People in the news

Posted on Mar 29, 2007

Eleven members of Campus Safety were sworn in as private college security officers in a recent ceremony at the Nott Memorial. They are: William H. Blanchard, Daniel M. Darling, Christopher M. Hayen, Richard M. McCrary, Keith G. McKenna, Patrick J. Morris, Gary S. Olsen, Michael J. Richards, William A. Sickinger, David M. Stern and Edward D. Teller. The officers completed 327 hours of additional training with instructors from the nearby Zone 5 Police Academy, giving them more authority to keep the campus community secure.

Binyavanga Wainaina, writer in residence, is among the finalists for the National Magazine Awards, the magazine industry's highest honor. Wainaina, of Kenya, recently was nominated in the fiction category, which honors the quality of a publication's literary selections. His piece, “Ships in High Transit,” was selected as part of the entry for The Virginia Quarterly Review. It had already won the literary journal's top short fiction prize for 2006. In February, a two-part interview with Wainaina aired on WAMC public radio, and the Albany Times Union ran a profile on him.

Teran R. Tadal '04, assistant dean of Admissions, was invited to join the board of Consortium for Educational Excellence through Partnerships (CEEP), the school-college partnership arm of Foundation for Excellent Schools (FES). She will help organize CEEP events and work to create stronger bonds between FES-affiliated schools and colleges.

Tarik Wareh, the John D. MacArthur Assistant Professor of Classics, will deliver a paper at the annual meeting of the Classical Association of the Middle West and South in Cincinnati, Ohio, April 14. The paper is titled, “School Politics and the Monarch's Court: Speusippus' Letter to Philip.”

## \$2 million gift means a new moniker for a Minerva

Posted on Mar 29, 2007

David J. Breazzano '78, co-founder and principal of an investment management firm in Waltham, Mass., has made a \$2 million gift to the College. In honor of the unrestricted gift, Orange House has been renamed Breazzano House, with an official dedication to be held ReUnion weekend, May 31-June 3.

“As one goes through life, it's clear you are defined by where you went to undergraduate school,” said Breazzano, a member of the College's Board of Trustees. “Everyone owes a substantial debt to their alma mater. My four years at Union were a critical part of my life, and I felt I needed to do something to repay that debt.”

At Union, Breazzano was president and treasurer of Phi Sigma Kappa and also got involved in intramural sports, radio station WRUC and the Interfraternity Council. He earned a B.A. in political science and economics, and an MBA in finance and accounting from Cornell University. He managed the Fidelity Capital & Income Fund and was chief investment officer of the T. Rowe Price Recovery Fund. In 1996, he and two partners founded DDJ Capital Management, which manages more than \$3 billion on behalf of some 80 institutional clients.

Breazzano's previous support to Union includes presidential scholarships and the expansion of Schaffer Library.

“David is a longtime College friend, and we are extremely grateful for his generous support and leadership,” said President Stephen C. Ainlay. “He has given freely to his alma mater through the years, and we are proud to consider him a part of our family.”

Breazzano has three sons, including Jeremy, a senior assigned to Breazzano House.