I wanted to share an amazing story of an apparently original discovery made by the TIAB students during one of our recent classes. While in the great scheme of mathematics, this discovery is but one new cute formula among thousands, it is a proof that anyone with enough curiosity can make new discoveries in math! What is most important to me is to continue nurturing the spirit of discovery among the children. At the end of the day, it does not matter if what they discover is truly original or has been found by someone else before. It is new to them and that is what matters! As long as they do not lose curiosity and love of math, the day will come when many of them will discover something truly amazing!
But back to the story of the discovery. About a month ago, our Level 2 group of students was studying the mysteries of the Fibonacci sequence and the infinitely beautiful patterns it contains. We were in the middle of talking about divisibility rules among Fibonacci numbers. The way I teach is this: instead of showing the patterns I lead the kids to discover the patterns for themselves. At the time, I presented to the kids Fibonacci sequence with every third, or every fourth, or every fifth number highlighted to help them find the divisibility pattern (see below). It was as if Fibonacci bunnies were playing the game of hopscotch.
Every 3rd number: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584
Every 4th number: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584
Every 5th number: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584
The pattern we were seeking is that in every row, all numbers highlighted in bold divide the first highlighted number and no other numbers do so. For example, in the first row, all bold numbers are even and all other numbers are odd. In the second row, all bold numbers are divisible by 3 and all other numbers are not. In formal mathematical terms, all Fibonacci numbers Fkn are divisible by Fn and no other Fibonacci numbers divide one another.
As we were studying the pattern, two students noticed another relationship between the highlighted numbers. It seemed to them that the ratio of the numbers highlighted in bold in each row was about the same. For example, in the first row, 8 divided by 2 is 4; 34 divided by 8 is just slightly greater than 4; 144 divided by 34 is again just slightly greater than 4; and so on. Similarly, in the second row, 21 divided by 3 is 7; 144 divided by 21 is just under 7; and so on.
While the ratio is not exact, it seemed persistent enough to explore further. As we looked more into it, the most beautiful relationship emerged. Let’s start with the first row. Here is the list of just every third Fibonacci number: 2 8 34 144 610 2584. And here is the relationship!
8 = 4 × 2
34 = 4 × 8 + 2
144 = 4 × 34 + 8
610 = 4 × 144 + 34
2584 = 4 × 610 + 144
Notice how the numbers in each column repeat themselves. You could do the same for every fourth, fifth, or n-th Fibonacci number and find the same pattern. Here it is in terms of the bunny hopscotch: “Consider a bunny hopping on every n-th Fibonacci number. Let the ratio of the first two such numbers be the bunny’s special coefficient. Then if you multiply this coefficient by any number a bunny hops on and either add or subtract the previous number she visits, you will find the next number she will hop on.”
For a more mathematically inclined reader, here is the formula in formal math terms:
F(k+1)n = (F2n / Fn) × Fkn + (-1)(n+1) × F(k-1)n
I must admit that while I read a lot about Fibonacci sequence, I have never encountered that pattern. When I came home that night, I first set out to prove that it always works. It took a bit of advanced algebra, and so unfortunately I cannot describe it in this post, but the pattern is legit!
I certainly do not consider myself to be a true expert in Fibonacci sequence. So, I wrote an e-mail to an expert – Dr. Arthur Benjamin, professor at Harvey Mudd College, who specializes in combinatorics and has done quite a bit of research in the field – and asked if he knew of the formula. I was very pleased to learn that it was also knew to Dr. Benjamin. Below is what Dr. Benjamin said:
“Congratulations to your students for making such an interesting discovery, and congratulations to you for inspiring them to make such discoveries. You and your students have my deepest respect.”
~ Dr. Arthur Benjamin