November's issue of The College Mathematics Journal features an article by three collaborators, Alif Anggoro, Eddy Liu, and Angus Tulloch, entitled "The Rascal Triangle." Here's the abstract: "A number triangle, discovered using a recurrence formula similar to that of Pascal’s triangle, yields sequence A077028 from the Online Encyclopedia of Integer Sequences."
Talk about burying the lede! The collaborators are middle-school students from the US, Canada, and Indonesia. The three middle-schoolers came up with an alternative way of populating the numbers in a triangular array. In Pascal's Triangle, you populate the triangle by adding the numbers directly above each number to the right and the left. This rule generates an infinite series of whole numbers, and the triangle has various uses in probability, binomial coefficients, and more.
According to their article, the boys were responding to an IQ test that had presented the numerical values of the first 5 lines of Pascal's Triangle, and asked what comes next. When they offered an "incorrect" answer, they set about defending their logic.
The Rascal Triangle proposes a different rule, which also generates an infinite series of whole numbers. I'll let Gary E. Davis, at the Republic of Math, illustrate the rule, since I'm not a mathematician. As Davis notes, "It is a lovely example of how much fun school mathematics can be if a teacher — in this case probably uncle Andy Liu — listens to students and encourages and nurtures their ideas."
In addition to the Republic of Math's writeup, you can also see the Mathematical Association of America's story for more information.
(My favorite GeekDad-related bit in the article: From Angus's bio: "I have to thank my parents for their steadfast refusal to buy a gaming system.")
