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The Rascal Triangle

Collaboration among mathematicians scattered across the globe isn’t especially noteworthy nowadays, except perhaps when the long-distance collaborators are middle school students who come up with a novel mathematical result.

Representing the United States, Canada, and Indonesia, middle schoolers Alif Anggoro, Eddy Liu, and Angus Tulloch now have not only a mathematical finding but also a published paper to their credit. Their article “The Rascal Triangle” appears in the November College Mathematics Journal (CMJ).

The young collaborators discovered their triangular number array using a recurrence formula similar to that for Pascal’s triangle. The formula yields a particular, known sequence of numbers, denoted A077028 in the On-Line Encyclopedia of Integer Sequences.

The first half of the paper is written as a dialogue between the authors and a teacher who is irritated that the authors respond with “1 4 5 4 1” (instead of “1 4 6 4 1”) to the request for the next row in the following array:

The students explain that they use the recurrence (West x East + 1)/North (in their picturesque terminology) instead of the standard Pascal recurrence (West + East) to obtain the numbers. The teacher challenges them to prove that their recurrence does produce an integer array, and to prove that their array is simpler than Pascal's. This they proceed to do.

Liu is an eighth-grader at Washington Middle School in Seattle, Washington. He served as the corresponding author for the paper. Tulloch is an eighth-grader at Crestomere School in Rimbey, Alberta. Anggoro, a seventh-grader at Al Azhar Junior High School in Java Bekasi, a suburb of Jakarta, Indonesia, has never met his fellow authors. Their collaboration was conducted via email.

Mathematician Andy Liu (University of Alberta) orchestrated the collaboration. Eddy Liu, Andy's nephew, first met Tulloch at a mathematics summer camp in Edmonton in 2008. "I met Alif in Manila during the First East Asian Mathematics Summer Camp in 2009," Andy Liu wrote in a recent email. "At that time, Eddy and Angus had been working on the triangle problem. However, like typical North America cars, their engines were not highly charged, but their brakes were mighty powerful. So I suggested to Alif to contact both Eddy and Angus, which he took the initiative to do so."

"I was completely charmed by the article from the start," said CMJ editor Michael Henle when asked why he decided to publish an article by such young authors. "It has the two essential elements of a good expository article for The College Mathematics Journal: a compelling story and interesting mathematics. The story is obviously the strongest part. Who hasn't imagined one-upping a teacher in this way? This story, especially its very effective dramatization in the article, provides the reader with a kind of vicarious revenge. 

"There is real mathematics here also: both the question of whether the numbers in the Rascal Triangle are integers and the question of its relative simplicity versus Pascal's triangle,” Henle said. “Some mathematicians might still question whether the article should be published, but the fact that the resulting sequence is listed in N. J. A. Sloane's On-Line Encyclopedia of Integer Sequences and that the diamond formula appears to be new made this decision a snap."

"The Rascal Triangle" (pdf)

The College Mathematics Journal is an expository journal aimed at teachers of college mathematics, particularly those teaching the first two years. Online and print subscriptions are available here.

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