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Middle School Students Publish New Mathematics: The Rascal Triangle

Consider this: Eddy Liu is an eighth-grader at Washington Middle School in Seattle, Washington; Angus Tulloch is an eighth-grader at Crestomere School in Rimbey, Alberta; and, Alif Anggoro is a seventh-grader at Al Azhar Junior High School in Java Bekasi, a suburb of Jakarta, Indonesia. Together, they did mathematical research through e-mail...and have just had their first results published...called the Rascal Triangle.

Think of the Pascal Triangle, as started with this picture:

By Pascal's rules, the next line of numbers would be 1 4 6 4 1.
The young trio of middle schoolers used this same start, but created a different triangular array of numbers using a different set of rules. The trio claimed that the next line of numbers was 1 4 5 4 1. Any ideas as to their generation rules?

Both approaches used the numbers directly above the desired number. In their middle-school terminology, Pascal used the rule [West + East], while the middle schoolers used the rule [West x East + 1]/North.

Their big task was to prove that their approach would always generate integers in a sequence. Can you prove this? And, what are the next three rows in the Rascal Triangle?

The result are published in the College Mathematics Journal (November 2010). Their rule does produce a previously known sequence of numbers, denoted A077028 in the On-Line Encyclopedia of Integer Sequences.

Though guided by Eddy Liu's uncle Any Liu, a mathematician at the University of Alberta, this collaboration by middle school students is impressive. When I was in middle school, I was still trying to figure out how to button my shirts in the right sequence.