“Some Really Cool Things Happening in Pascal’s Triangle” Jim Olsen Western Illinois University This version is compiled for Math 406.

Outline Triangular Numbers, Initial Characterizations of the elements of Pascal’s Triangle, other Figurate Numbers, and Tetrahedral numbers. Tower of Hanoi Connections in Pascal’s Triangle Catalan numbers in Pascal’s Triangle

1. Triangular numbers

In General, there are Polygonal Numbers Or Figurate Numbers Example: The pentagonal numbers are 1, 5, 12, 22, …

Let’s Build the 9th Triangular Number 1 Let’s Build the 9th Triangular Number +2 +3 +4 +5 +6 +7 +8 +9

Each Triangle has n(n+1)/2 Take half. Each Triangle has n(n+1)/2 n n+1

Another Cool Thing about Triangular Numbers Put any triangular number together with the next bigger (or next smaller). And you get a Square!

Some Basic Characterizations of Pascal’s Triangle First Definition: Get each number in a row by adding the two numbers diagonally above it (and begin and end each row with 1).

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. etc. Five Choose Two

{A, B} {A, B} {A, C} {A, C} {A, D} {A, D} {A, B, C, D, E} etc. etc. Form subsets of size Two  Five Choose Two

Therefore, the number of combinations of a certain size is the same as the number of subsets of that size.

Characterization #1 and characterization #2 are equivalent, because

Characterization #3 Symmetry or “Now you have it, now you don’t.”

Characterization #4 Why? The total of row n = the Total Number of Subsets (from a set of size n) =2n 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.

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. 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. 1 + 5x4 + T4x3 1+20+30 = 51

The kth n-gonal Number is: Polygonal numbers always begin with 1. 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. 1 + (k-1)x(n-1) + Tk-2x(n-2)

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. 12 Days of Christmas Why? Because we use the Hockey Stick Principle to sum up triangular numbers.

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 a 2 3 aba 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 #12 The 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:

Look at the Sequence as the disks Moves Needed Sequence 2 3 aba

Look at the Sequence as the 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 Hanoi Can you see the ruler markings? Disks Moves Needed Sequence 1 a 2 3 aba 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 Solution to Tower of Hanoi

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 #13 If 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 Solution to Tower of Hanoi Sierpinski Gasket/Wire Frame

…But isn’t all of this Yes/No…..On/off Binary Base Two

Characterization #12.1 Count in Base-2 The 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. 1 Digit 2 Digits 3 10 11 100 101 110 111 Count in Base-2

1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 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 3 Digits 1 10 11 100 101 110 111

Solution to Tower of Hanoi Sierpinski Gasket/ Wire Frame Ruler Markings 1 10 11 100 101 110 111 Solution to Tower of Hanoi Sierpinski Gasket/ Wire Frame Binary Numbers

Catalan numbers are also in Pascal’s Triangle 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)

What are the categories? 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?

Number of Toothpicks from each 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: 3432 But…. This double counts, so there are 1716 toothpicks!

Western Illinois University Thank You 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