May 2002 issue of FOCUS
Math Circle students discovered a new, innovative and elegant approach to a classic problem in combinatorial geometry. An article about their findings appears in the May 2002 issue of FOCUS, The Mathematical Association of America's largest circulating periodical.
Here's the opening section ....
Young Students Approach Integer Triangles
By James Tanton and David Batuner, Alexander Belyi, Owen Callen, John Coyne, Adam Donovan, Rostic Gorbatov, Avi Levitan, Sam Lichtenstein, Alan McAvinney, Benjamin Moody, Peter Sergey Panov, David Plotkin, Ben Plotkin-Swing, Yuri Podpaly, Noah Rosenblum, Teresa Shirkova, Sonya Shteingold, Matthew Tai.
How many different triangles of perimeter n can be made with a supply of n toothpicks? Let us assume all triangles created have an integral number of toothpicks per side and have positive area. We wish to count the number of incongruent triangles that can be so constructed. This is the problem students aged 12 - 17 grappled with over ten one-hour sessions of The Math Circle in the fall of 2001.
Organized as a school for the enjoyment of mathematics, The Math Circle offers programs of courses for young students, aged 5 through 18, who enjoy math and want more than the typical school curriculum offers. We hold classes at Harvard University and Northeastern University, and at community education centers. The school was founded by Robert Kaplan, Ellen Kaplan and Tomás Guillermo in 1994, and I have been fortunate to be involved with this program for the past two years.
We typically begin a semester by posing a mathematical problem to each of our classes and allowing the small groups of students to explore the mystery for the ten weeks that follow. The role of an instructor in The Math Circle is not so much to instruct as to guide and offer occasional hand-holds (or sometimes to throw spanners into the works). We want our students to experience for themselves the creative aspects of mathematics, to make mistakes, to experience the frustrations of doing math, as well as the deep joy when inspiration finally arrives. In short, we want our students to own the mathematics they encounter.
With all boundaries removed it is amazing just how far young students can, and do, go. This was shown by the accomplishments of my students in designing a novel solution to the toothpick triangle problem. It was a collaborative effort. The point of The Math Circle is not to foster competition but to allow, and encourage, students to take intellectual risks. The setting is collegial and relaxed. With free discussion of ideas and play of invention, innovation can blossom. The toothpick triangle problem, like all of our problems, has the added advantage of being immediately accessible and mathematically rich.
Trial and error shows that a supply of 13 toothpicks can produce five incongruent triangles. We denote them as triples: (6, 6,1), (6, 5, 2), (6, 4, 3), (5, 5, 3) and (5, 4, 4). There is a surprise if you add one toothpick: the count decreases. Only four integer triangles can be made with 14 toothpicks. Let T(n) denote the number of possible triangles for a given perimeter n (for n in N). T(13)=5 and T(14)=4 shows that the function T(n) is certainly not monotonic. Is there a general formula for it? My students didn't know that this problem is well known and complete specification of the function T(n) has already been established (see ,  and also [2, ch. 3],  and [6, ch. 6]). The approaches taken in the literature make use of algebraic machinery that can only be described as hefty: generating functions, multiple-term partial fractions involving cubics, and the binomial theorem to extract coefficients (though see  for an alternative direct approach). I was astonished to see my students solve this problem completely using nothing more than formulas for quadratics.
This paper describes the evolution of their approach. They began classically by first discovering a connection to partition functions (and they made some new observations along the way). Thereafter, their method is surprisingly simple, elegant, and new!