Middle School Students Publish New Mathematics: The Rascal Triangle
Consider this: Eddy Liu is an eighthgrader at Washington Middle School in Seattle, Washington; Angus Tulloch is an eighthgrader at Crestomere School in Rimbey, Alberta; and, Alif Anggoro is a seventhgrader at Al Azhar Junior High School in Java Bekasi, a suburb of Jakarta, Indonesia. Together, they did mathematical research through email...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 middleschool 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 OnLine 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.
