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“Some Really Cool Things Happening in Pascal’s
Triangle”
Jim Olsen
Western Illinois University
This version is compiled for Math 406.
Outline
1. Triangular Numbers, Initial Characterizations of the elements of Pascal’s Triangle, other Figurate Numbers, and Tetrahedral numbers.
2. Tower of Hanoi Connections in Pascal’s Triangle
3. Catalan numbers in Pascal’s Triangle
1. Triangular numbers
15 10 6 3 1 54321 TTTTT
In General, there are Polygonal Numbers
Or Figurate Numbers
Example: The pentagonal numbers are
1, 5, 12, 22, …
+31+2 +9+8+7+6+5+4
Let’s Build the 9th
Triangular Number
459 T
n
n+1
n(n+1)
Take half.
Each Triangle
has n(n+1)/2
2
)1(
nnTn
Another Cool Thing about Triangular Numbers
Put any triangular number together with the next bigger (or next smaller).
21 nTT nn
And you get a Square!
819298 TT
Characterization #1
• First Definition: Get each number in a row by adding the two numbers diagonally above it (and begin and end each row with 1).
Some Basic Characterizations of Pascal’s Triangle
Example: To get the 5th element in row #7, you add the 4th and 5th element in row #6.
Characterization #2
• Second Definition: A Table of Combinations or Numbers of Subsets
• But why would the number of combinations be the same as the number of subsets?
etc.
Five Choose Two
2
5
etc.
2
5
{A, B} {A, B}
{A, C} {A, C}
{A, D} {A, D}
etc.
etc.
Form subsets of size Two Five Choose Two
{A, B, C, D, E}
• Therefore, the number of combinations of a certain size is the same as the number of subsets of that size.
subsets 10
102
5 2 choose 5
subsets 120
1207
10 7 choose 10
Characterization #1 and characterization #2 are equivalent, because
r
n
r
n
r
n 1
1
1
Characterization #3
Symmetry or“Now you have it, now you don’t.”
7
9
2
9
rn
n
r
n
Characterization #4
The total of row n
= the Total Number of Subsets (from a set of size n)
=2n
32215101051 5
n
n
nnnn2
210
Why?
Characterization #5
The Hockey Stick Principle
The Hockey Stick Principle
Characterization #6
The first diagonal are the “stick” numbers.
…boring, but a lead-in to…
Characterization #7
The second diagonal are the triangular numbers.
Why?
Because of summing up stick numbers and the Hockey Stick Principle
Triangular Number Properties
Relationships between Triangular and Hexagonal Numbers….decompose a hexagonal number into 4 triangular numbers.
Notation
Tn = nth Triangular number
Hn = nth Hexagonal number
Decompose a hexagonal number into 4 triangular numbers.
)34(...51 nH n
nTn ...21
12
13
nn
nnn
TH
TTH
)12( nnH n
A Neat Method to Find Any Figurate Number
Number example:
Let’s find the 6th pentagonal number.
The 6th Pentagonal Number is:
• Polygonal numbers always begin with 1.
1 + 5x4 + T4x3 1+20+30 = 51
• Now look at the “Sticks.”
– There are 4 sticks
– and they are 5 long.
• Now look at the triangles!
– There are 3 triangles.
– and they are 4 high.
The kth n-gonal Number is:
• Polygonal numbers always begin with 1.
1 + (k-1)x(n-1) + Tk-2x(n-2)
• Now look at the “Sticks.”
– There are n-1 sticks
– and they are k-1 long.
• Now look at the triangles!
– There are n-2 triangles.
– and they are k-2 high.
Now let’s add up triangular numbers (use the hockey stick principle)….
A Tetrahedron.
And we get, the 12 Days of Christmas.
Characterization #8
The third diagonal are the tetrahedral numbers.
Why?Because we use the Hockey Stick Principle
to sum up triangular numbers.
12 Days of Christmas
Formula for the nth tetrahedral number
…see it…
From
Proofs
Without
Words
Characterization #9
Pascal’s triangle is actually a table of permutations.
Permutations with repetitions. Two types of objects that need to be arranged.
For Example, let’s say we have 2 Red tiles and 3 Blue tiles and we want
to arrange all 5 tiles.
There are 10 permutations.Note that this is also 5 choose 2.Why?Because to arrange the tiles, you
need to choose 2 places for the red tiles (and fill in the rest).
Or, by symmetry?…
2. Characterizations involving Tower of Hanoi, Sierpinski, and _______ and _______.
• Solve Tower of Hanoi.
• What do we know? Brainstorm.• http://www.mazeworks.com/hanoi/index.htm
Solutions to Tower of Hanoi
Disks Moves Needed
Sequence
1 1 a
2 3 aba
3 7 aba c aba
4 15 aba c aba D aba c aba
5 31 aba c aba D aba c aba E aba c aba D aba c aba
Characterization #12The sum of the first n rows of Pascal’s Triangle
(which are rows 0 to n-1) is the number of moves needed to move n disks from one peg to another in the Tower of Hanoi.
Notes:
• The sum of the first n rows of Pascal’s Triangle (which are rows 0 to n-1) is one less than the sum of the nth row. (by Char.#4)
• Equivalently: 122...222 1210 nn
Look at the Sequence as the disks
Disks Moves Needed
Sequence
2 3 aba
Look at the Sequence as the disks
Disks Moves Needed
Sequence
3 7 aba c aba
What does it look like?
Look at the Sequence as the disks
A ruler!
Solutions to Tower of HanoiCan you see the ruler markings?
Disks Moves Needed
Sequence
1 1 a
2 3 aba
3 7 aba c aba
4 15 aba c aba D aba c aba
5 31 aba c aba D aba c aba E aba c aba D aba c aba
Solution to Tower of Hanoi
Ruler Markings
What is Sierpinski’s Gasket?
http://www.shodor.org/interactivate/activities/gasket/
It is a fractal because it is self-similar.
More Sierpinski Gasket/Triangle Applets and Graphics
http://howdyyall.com/Triangles/ShowFrame/ShowGif.cfm
http://www.arcytech.org/java/fractals/sierpinski.shtml
by Paul Bourke
Vladimir Litt's, seventh grade pre-algebra class from Pacoima Middle School Pacoima,
California created the most amazing Sierpinski Triangle.
http://math.rice.edu/%7Elanius/frac/pacoima.html
Characterization #13If you color the odd numbers red and the even
numbers black in Pascal’s Triangle, you get a (red) Sierpinski Gasket.
http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html
Solution to Tower of Hanoi
Sierpinski Gasket/Wire Frame
Ruler Markings
…But isn’t all of this
• Yes/No…..On/off
• Binary
• Base Two
Characterization #12.1The sum of the first n rows of Pascal’s Triangle
(which are rows 0 to n-1) is the number of non-zero base-2 numbers with n digits.
1Digit
2Digits
3Digits
1 11011
11011
100101110111
Count in
Base-2
11011
100101110111
10001001101010111100110111101111
What Patterns Do You See?
How can this list be used to solve Tower of Hanoi?
Binary Number List Solves Hanoi
Using the list of non-zero base-2 numbers with n digits. When:
• The 20 (rightmost) number changes to a 1, move disk a (smallest disk).
• The 21 number changes to a 1, move disk b (second smallest disk).
• The 22 number changes to a 1, move disk c (third smallest disk).
• Etc.
a b a C a b a
3Digits
1 10 11
100101110111
Solution to Tower of Hanoi
Sierpinski Gasket/ Wire Frame
1 10 11
100101110111
Binary Numbers
Ruler Markings
The Catalan numbers are a sequence of natural numbers that occur in numerous counting problems, often involving recursively defined objects.
They are named for the Belgian mathematician Eugène Charles Catalan (1814–1894).
The first Catalan numbers 1, 1, 2, 5, 14, 42,132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845…
(“numerous” is an understatement)
Catalan numbers are also in Pascal’s Triangle
A Fun Way to Count the Toothpicks in the 12 Days of Christmas Tetrahedron
Organize the marshmallows (nodes) into categories, by the number of toothpicks coming out of the marshmallow.
What are the categories?
This double counts, so there are 1716 toothpicks!
Category of Nodes
Number of Nodes
Number of Toothpicks from each
Product
Corners 4 3 12
Edges 6x10 6 360
Faces 4xT9 9 1620
Interior Te8 12 1440
Total: 3432But….
Jim Olsen
Western Illinois University
www.wiu.edu/users/mfjro1/wiu/index.htm
www.wiu.edu/users/mfjro1/wiu/tea/pascal-tri.htm
Thank You