A Beautiful Mind
A few weeks ago Koren Burling, a science teacher at Spring Creek Elementary, sent out an email soliciting judges for this year’s Science Fair for 5th and 6th graders. My daughter recently became a third grader at Spring Creek, so I was only too happy to help. I was intrigued by a comment Koren made in her email. One of the students had done a project on the Fibonacci numbers, and Koren was especially keen to find judges that would be able to understand that project. What 5th or 6th grader is interested in Fibonacci numbers, I thought. This turned out to be my introduction to a very special young lady.
She was the last student I judged today, which is a good thing for the other students, for their projects would have suffered from the comparison. When I walked up to her display, I knew immediately I was in for a treat. The science teacher preceded me and told a strikingly beautiful young girl that she had found a judge. The little girl did what all little girls do when excited–she pumped her fists and squealed in delight. I didn’t even have time to look at her display before she launched into her exposition.
In quick strokes she told me the story of Fibonacci and his numbers. She let me know how quickly they grew, how they could be derived from a simple rule, and how they contained many magical patterns. Then she dove into the details of one such pattern, that every fifth Fibonacci number is divisible by 5. She didn’t just tell me this, she showed me an argument–”also known as a proof,” she added–that explained why it was so. The argument was exquisite, an elegant recursion based on an unusual recurrence for the Fibonacci sequence:
F(n) = 5*F(n-6) + 8*F(n-5)
You can see from this that the result is true by induction (though she did not use the term). It is certainly the case the F(5) is divisible by 5, since F(5) is equal to 5. I.e., the first five Fibonacci numbers are 1, 1, 2, 3, 5. Now the term 5*F(n-6) is divisible by 5, so we need only worry about the second term, 8*F(n-5). This is divisible by 5 precisely when F(n-5) is divisible by 5. Well then, F(5) is divisible by 5, so F(10) is divisible by 5, so F(15) is divisible by 5, and so on for all multiples of 5.
Not bad, I thought for a 5th or 6th grader. Even better for a 4th grader, which she was according to my judging sheet. Wow. Only a handful of 4th graders participated in the fair, and this was obviously a special one. She understood so much about the Fibonacci sequence, and she was so knowledgeable about it that I couldn’t resist. I asked her if she knew why her alternative equation for the Fibonacci numbers was true. She did not, so I decided to teach her. Start by assuming the formula holds for values less than n. In particular,
F(n-1) = 5*F(n-7) + 8*F(n-6) F(n-2) = 5*F(n-8) + 8*F(n-7)
Using the standard definition of the Fibonacci sequence, we have that
F(n) = F(n-1) + F(n-2)
= 5*F(n-7) + 8*F(n-6) + 5*F(n-8) + 8*F(n-7)
From here, we should collect terms and simplify. But that requires more algebra than even a very smart 4th grader can handle. So I chose a different approach. Let’s start from the other end. We want
F(n) = 5*F(n-6) + 8*F(n-5)
So we simplify the right hand as follows:
5*F(n-6) + 8*F(n-5) = 5*(F(n-7) + F(n-8)) + 8*(F(n-6) + F(n-7))
= 5*F(n-7) + 5*F(n-8) + 8*F(n-6) + 8*F(n-7)
Now, this sum is exactly the same as our previous sum for F(n). So the formula holds.
Well, that’s good enough for most people, and I left it at that. Although there is still a minor problem. We have yet to establish that the formula holds in the base case. I’ll leave that as an exercise to the reader, noting only that it is only necessary to determine the formula holds for the 7th Fibonacci number. And it does.
Many 4th graders would have been bored by this exposition. Others would have found it impossible to follow. But this precocious kid was neither bored nor mystified. She loved it, and she could follow the argument just fine. In fact, it was she who simplified some of the algebra to make the result come out. This was obviously a special kid, and I decided I just had to remember her name. So I looked at her nametag and found “Sarah Shader.” Shader, I thought? That name seems a bit familiar.
Then it hit me! Bryan Shader is one of the professors in the math department. In fact, he and his wife are both professors in the math department, and so was his father. Mathematics runs in her family. I had to laugh. This was just too amazing. So I congratulated her on her work, encouraged her as much as I could, and gave her a glowing written report. It was the least I could do. Because truly, I was the one who was blessed, to have spent a little time with such a beautiful mind.
Now for the next big step. How do I get my own little girl—whom I consider to be just as smart—and this little girl together? I know a little something about growing up a prodigy, and one thing I remember is that prodigies love company, but only smart company. I’m sure these two girls could grow up best of friends.