Student's article published
Laura McCormick, LSUS math major (and president of the LSUS Math and Physics Club), has an article published in the November 2009 issue of the Gazette of the Australian Mathematical Society.
The article, Square products of punctured sequences of factorials, is co-authored with Dr. Rick Mabry of the LSUS Department of Mathematics. Laura presented the article's results at the Texas Undergraduate Mathematics Conference at Sam Houston State University, November 7, 2009, and again in a poster session at the Nebraska Conference for Undergraduate Women in Mathematics, Jan. 29-31, 2010.
The story behind the research is inspiring. McCormick was a student in Mabry's second semester calculus course (MATH 222) in the spring of 2009 when her professor described to the class a problem that appeared in one of the popular math journals. "I showed them something I had recently proved about infinite series," he recalls, "a topic we happened to be studying at the time." He also described the process of solving the problems and submitting solutions, and the fact that a solver could get his or her name printed in the journal. "I usually mention that the journal editors are particularly fond of printing solutions by students, since that can be a big boost and inspiration to an early career," says Mabry. "If I remember correctly, Laura approached me after class and said something like, 'I want to prove something, too!'"
"I would not have said something so geeky," McCormick quips. But even if those weren't the exact words, the exchange set a big wave in motion.
Dr. Mabry gave Laura a list of some journals available in the LSUS library and online, and suggested she browse for a current problem that she found appealing. Within a day or two, Laura had chosen a problem and visited Dr. Mabry's office to get some advice. "I had a somewhat negative reaction to her selection," Mabry says. "I think I told her that that kind of problem hurts my head and that she was on her own! Maybe I shouldn't have said that." Or maybe it was just the right response. Mabry came to Bronson Hall one or two mornings later to find Laura in front of a chalkboard full of numbers. It was Laura's solution to the problem.
The professor sat, checked the work, and found it correct. "And it did hurt my head," says Mabry, "partly because it was a very long solution. But it is natural for a student's first attempts to be lacking in what is called 'elegance' by mathematicians, so I thought it would help to get her solution published if we could streamline the proof."
The solution did not satisfy Mabry for other reasons. "Often a mathematical fact can be proven in such a way that it does not really reveal what is going on. This "puzzle" was an isolated case of a more general problem. I started wondering, and I did a quick experiment..."
But first, here is the original puzzle, titled "Factorial fun" in the journal.
The numbers 1!, 2!, 3!, . . . , 100! are written on a blackboard. Is it possible to erase one of the numbers so that the product of the remaining 99 numbers is a perfect square?
(Math students will recognize the factorial notation; for example, 5!=(1)(2)(3)(4)(5)=120.)
Laura discovered that by erasing the number 50!, the answer was "Yes."
"I don't know how she figured it out," Mabry relates, "I did it using the computer!" Referring to the experiment mentioned above, he explains, "I wanted to know if the choice of 50! was simply because 50=100/2." Mabry and McCormick put it this way in their article:
"Upon seeing the answer above, one is immediately tempted to make this conjecture: with arbitrary even n instead of 100, the answer will be ‘Yes, erase (n/2)!'. But this is false, since for n = 98, we have to ‘erase' the number 50!, not 49!, while for n = 102, no such erasable number exists! Something funny is going on here."
The power of the computer was not lost on Laura. "It kinda blew me away," she says. "Dr. Mabry just hopped up and typed a few lines into the computer and we immediately had all sorts of new conjectures to prove. At first I thought, what have I gotten myself into? I thought I was already finished! But in truth, I was hooked." Within a week or two they solved a very general version of the problem.
"So not only did we submit a solution to the original problem, which is all I had hoped for, we actually proved an original theorem and got a separate paper published," says McCormick. "I started to see the computer as an experimental tool that could lead to discovery. I also learned that mathematical writing can be fun. It doesn't have to be just dry equations. The article is actually sort of a hoot. And it all happened so fast, in one semester. I have to say that it changed my life."
And that is part of a much longer story. A few short years later, as of Aug. 2011, Laura is a teaching assistant and graduate student in the math Ph.D. program at the University of South Carolina. "What have I done?" asks Mabry, half-joking. "When I met her, Laura was intent on a career in engineering. She was very serious about it and had already been checking into engineering graduate programs. " But the world needs mathematicians, too, as the professor is quick to point out. Not only that, but "the mathematical life is a good life, a creative life. We have fun almost every day."
Factorial fun, indeed! In fact, the fun is not over. Laura later proved results about cubes and higher powers, and noticed that the techniques they used ran out of gas at the power 7. "So maybe seven isn't always so lucky," Mabry muses, though he believes the results already obtained might be publishable. "But we'll try to find a way around the impediment. Every stumble, every difficulty, is an opportunity for a new discovery."
- The original puzzle: Puzzle Corner 11, Gaz. Aust. Math. Soc., 36, no. 1 (March 2009), 12-16. (See p. 13.)
- The puzzle solution by McCormick and Mabry: Puzzle Corner 13, Gaz. Aust. Math. Soc., 36, no. 3 (July 2009), 176-179. (See p. 178.)
- The article: Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 36, no. 5 (Nov. 2009), 346-352.