“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

jr-olsen@wiu.edu

www.wiu.edu/users/mfjro1/wiu/index.htm

www.wiu.edu/users/mfjro1/wiu/tea/pascal-tri.htm

Thank You