SOME FIBONACCI SURPRISES
The Power of Visualization
James Tanton
MAA Mathematician-at-Large
tanton.math@gmail.com
Curriculum Inspirations: www.maa.org/ci
Mathematical Stuff: www.jamestanton.com
Mathematical Courses: www.gdaymath.com
1  2  3  ...  N 
N  N  1
2
Visualization in the curriculum
* “Visual” or “Visualization” appears 34 times in the ninety-three
pages of the U.S. Common Cores State Standards
- 22 times in reference to grade 2-6 students using visual models
for fractions
- 1 time in grade 2 re comparing shapes
- 5 times re representing data in statistics and modeling
- 4 times re graphing functions and interpreting features of graphs
- 2 times in geometry re visualizing relationships between two- and
three-dimensional objects.
* Alberta curriculum: Recognised HS core mathematical process:
[V] Visualization “involves thinking in pictures and images, and the
ability to perceive, transform and recreate different aspects of the
world” (Armstrong, 1993, p. 10). The use of visualization in the
study of mathematics provides students with opportunities to
understand mathematical concepts and make connections among
them.
The Fibonacci numbers…
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
… arise in a myriad of contexts.
For example, count the number of sequences of Os and Xs of a given
length, avoiding two consecutive Xs.
Go the other extreme: Insist that Xs come in pairs!
Count the number of ordered partitions of a given integer that avoid 1:
Or count the number of ordered partitions that use two different types of 1!
Count the number of ways to arrange non-nested parentheses around
a string of objects:
Count the number of ways to stack (two-dimensional) cannon balls so that each
row is contiguous.
OR …
The language of ABEEBA uses only three letters of the alphabet: A, B, and E.
* No word begins with an E.
* No word has the letter E immediately following an A.
* All other combinations of letters are words.
Actually, words that begin with an E are allowed. They are swear words.
Let’s count the swear words in the language of ABEEBA.
And so on!
There is one visual model that explains
all these examples - and so much more!
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
The Fibonacci numbers arise from a classic honeycomb path-counting puzzle:
We see that the number of paths from the left cell to the Nth cell of the
honeycomb is the Nth Fibonacci number FN.
Actually … the number of paths between any two cells N cells apart is FN .
LET’S NOW HAVE SOME FUN!
THE BASIC APPROACH
Look at all sorts of features of a path between cells.
And so on.
Here’s a path between two cells on the top row.
The path “touches down” one groups of dots in the bottom row.
This matches placing non-nested parentheses around those dots.
And conversely … A set of non-nested parentheses determines a path.
Each step in a path either follows a diagonal step or skips over
two diagonal steps. We see a partition of a number into 1s and 2s.
N diagonals corresponds to a path to the N+1 th cell.
Count the ordered partitions of a number with two different types of 1.
1=1
2 partitions
2 = 1+1 = 1+1 = 1+1 = 1+1
5 partitions
3
= 2+1 = 2+1 = 1+2 = 1+2
= 1+1+1 = 1+1+1 = 1+1+1 = 1+1+1 = 1+1+1 = 1+1+1 = 1+1+1 = 1+1+1
Indeed every second Fibonacci number.
13 partitions
Each diagonal step in a path is a “break” between dots.
1 +1 + 3 +1 + 5
+ 1 = 12
There are an odd number of dots between breaks.
EXAMPLE: 5 = 3+1+1 = 1+3+1 = 3+1+1
= 1+1+1+1+1
There are five odd partitions of five.
Focus on the dots that a path misses.
OXOOXOXOOX
Missed circles define the path.
The first dot is never missed, the last dot is never missed,
no two consecutive dots along the zigzag are ever missed.
Ignoring end dots, draw Os on the dots hit and Xs on the dots missed.
Get sequence of Os and Xs along the zigzag avoiding two consecutive Xs.
Draw extra line segments at the beginning and end of the zigzag.
Each dot a path misses breaks the zigzag line of segments:
No two consecutive dots circled  No section just 1 segment long.
So we have a one-free partition of the number of the zig-zag steps.
EXAMPLE: 5 = 2+3 = 3+2
There are three one-free
partitions of five.
The language of ABEEBA.
Consider paths between cells on the top row.
Cannonball stacks:
Each stack gives a sequence of diagonal and horizontal steps…
… which gives a path between two cells on the top row.
Another approach to cannonballs:
There are three ways to make a stack with an extra row:
* add a ball to the left of a previously made stack
* add a ball to the right of a previously made stack
* place a previously made stack on top of a next row
But there is double counting.
But we saw today
We have the identity:
PRODUCTS OF PARTITIONS
(Inspired by a conversation with Sam Vandervelde)
Take all partitions on N, multiply terms, and add.
ALWAYS FIBONACCI?
Consider paths that end on a lower cell. Consider all possible locations of the UP steps.
12 dots on top row
24 dots in all
Single DOWN in each section.
There 2x4x1x2x3 ways to place the DOWNs.
A partition of 12 with terms multiplied together.
The sum of all such products counts all paths.
Answer must be F24 .
TRIANGULAR SUMS
Consider every second Fibonacci number: 1, 2, 5, 13, 34, 89, …
SUM ALWAYS ONE LESS THAN A FIBONACCI NUMBER?
Consider paths that end on a particular top dot, 2N + 1.
(N+1 dots on top row, N dots on bottom row.)
1 path
N paths touch just 1 lower dot
(N-k) places for span of k dots.
F2k+1 paths using those dots.
(N-k) x F 2k+1 paths touch a span of k dots.
This accounts for all paths:
1  N  F1   N  1 F3 
 ( N  k ) F2k 1 
 1 F2 N 1  F2 N 1
FIBONACCI IDENTITIES
Take your favourite Fibonacci identity and try to prove via paths.
EXAMPLE:
PROOF:
Here is my favourite identity:
EXAMPLE: 12 is divisible by 2, 3, 4 and 6, and
F12= 144 is divisible by F2 = 1, F = 2, F4 = 3 and F = 8.
3
I’ve always wondered …
Is there a formula for the quotient?
6
What if there is a remainder?
So in full generality …
PROOF:
There are a myriad of Fibonacci identities that perhaps can be proved via
path walking. (Care to try?)
CHALLENGE FOR TODAY:
Weeks of fun to be had all with the POWER OF A PICTURE!
THANK YOU!
[ I posted this very PowerPoint presentation on the front page of
www.jamestanton.com
]
SOME FIBONACCI SURPRISES
The Power of Visualization
James Tanton
MAA Mathematician-at-Large
tanton.math@gmail.com
Curriculum Inspirations: www.maa.org/ci
Mathematical Stuff: www.jamestanton.com
Mathematical Courses: www.gdaymath.com