“Some Really Cool Things Happening in Pascal’s Triangle” Jim Olsen Western Illinois University

Outline 0. What kind of session will this be? Triangular numbers Eleven Cool Things About Pascal’s Triangle Tetrahedral numbers and the Twelve Days of Christmas Solve Two Classic Problems. A Neat Method to find any Figurate Number

0. What kind of session will this be? This session will be less like your typical teacher in-service workshop or math class, and more like A play A musical performance Sermon Trip to a Museum I’m going to move quickly. I will continually explain things at various levels.

Answering non-Math Questions in advance Q: How does this relate to my classroom, NCTM, AMATYC, ILS, NCLB, …? A1: Understand it first, then get creative. A2: Get the handouts and websites at the back. A3: Communicate with me. I’d like to explore more answers to this question.

Answering non-Math Questions in advance Q: Isn’t this just math trivia? A: No. These are useful mathematical ideas (that are “deep,” but not hard to grasp) that can be used to solve many problems. In particular, basic number sense problems, probability problems and problems in computer science.

What kind of session will this be? I want to look at the beauty of Pascal’s Triangle and improve understanding by making connections using various representations.

2. 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

Interesting facts about Triangular Numbers Number of People in the Room Number of Handshakes 2 1 3 4 6 5 10 15 The Triangular Numbers are the Handshake Numbers Which are the number of sides and diagonals of an n-gon.

A B E D C Number of Handshakes = Number of sides and diagonals of an n-gon. A B E D C

Why are the handshake numbers Triangular? Let’s say we have 5 people: A, B, C, D, E. Here are the handshakes: A-B A-C A-D A-E B-C B-D B-E C-D C-E D-E It’s a Triangle !

Q: Is there some easy way to get these numbers? A: Yes, take two copies of any triangular number and put them together…..with multi-link cubes.

9x10 = 90 Take half. Each Triangle has 45. 9 9+1=10

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!

Eleven Cool Things About Pascal’s Triangle

Characterization #1 First Definition: Get each number in a row from 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 For example, Show me the proof

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 we use the Hockey Stick Principle to sum up stick numbers.

Now let’s add up triangular numbers (use the hockey stick principle)…. And we get, the 12 Days of Christmas. A Tetrahedron.

Characterization #8 The third diagonal are the tetrahedral numbers. Why? Because we use the Hockey Stick Principle to sum up triangular numbers.

Tetrahedral Numbers are Cool Like Triangular Numbers Do the same things. Find a general formula. Add up consecutive Tetrahedral Numbers.

Find a general formula. Use Six copies of the tetrahedron !

Combine Two Consecutive Tetrahedrals You get a pyramid! Wow, which is the sum of squares. (left for you to investigate)

Characterization #9 This 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. How many permutations (arrangements) are there?

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?…(

Characterization #10 Imagine a pin at each location in the first n rows of Pascal’s Triangle (row #0 to #n-1). Imagine a ball being dropped from the top. At each pin the ball will go left or right. ** The numbers in row n are the number of different ways a ball being dropped from the top can get to that location. Row 7 >> 1 7 21 35 35 21 7 1

Ball dropping There are 21 different ways for the ball to drop through 7 rows of pins and end up in position 2. Why? Because position 2 is And the dropping ball got to that position by choosing to go right 2 times (and the rest left).

Pascal’s Triangle gives it to you for any size grid! Equivalently To get to Wal-Mart you have to go North 2 blocks and East 5 blocks through a grid of square blocks. There are 7 choose 2 (or 7 choose 5) ways to get to Wal-Mart. Pascal’s Triangle gives it to you for any size grid!

Characterization #11 The fourth diagonal lists the number of quadrilaterals formed by n points on a circle.

The fourth diagonal lists the number of quadrilaterals formed by n points on a circle. Why? Because to get a quadrilateral you have to pick 4. Note that each quadrilateral has two diagonals and hence contributes one point of intersection in the interior of the n-gon.

Characterization #11 note The fourth diagonal lists the number of quadrilaterals formed by n points on a circle. The fourth diagonal lists the number intersection points of diagonals (in the interior) of an n-gon. =

Now to Solve Two Classic Problems If you connect n random points on a circle, how many regions do you get? (What is the most number of regions?) If you cut a pizza with n random cuts, how many regions do you get? (What is the most number of regions?)

If you connect n random points on a circle, how many regions do you get?

The number of regions in a circle (or pizza) with cuts Number of Regions equals 1 plus Number of Lines plus Number of Intersection points. (see the article by E. Maier, January 1988 Mathematics Teacher)

Answer to: If you connect n random points on a circle, how many regions do you get?

Answer to: If you connect n random points on a circle, how many regions do you get?

2. If you cut a pizza with n random cuts, how many regions do you get? Answer to: 2. If you cut a pizza with n random cuts, how many regions do you get?

If you cut a pizza with n random cuts, how many regions do you get? Answer to: If you cut a pizza with n random cuts, how many regions do you get? Example: 6 cuts in a pizza give a maximum of 22 pieces.

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)

Western Illinois University jr-olsen@wiu.edu Resources, Thank you. Jim Olsen Western Illinois University jr-olsen@wiu.edu faculty.wiu.edu/JR-Olsen/wiu/

First we will show this for a number example: Addendum We wish to show the following: First we will show this for a number example: n = 5; r = 3. Therefore, we wish to show:

To show We will build the subsets from the subsets and subsets

The 4 choose 2 subsets are subsets of size 2 from the pool {1, 2, 3, 4}. The 4 choose 2 subsets are: The 4 choose 2 subsets become: {1,2} {1,3} {1,4} {3,4} {2,4} {2,3} {1,2,5} 5 {1,3,5} 5 {1,4,5} 5 {2,3,5} 5 {2,4,5} 5 {3,4,5} 5 This gives us some of the 5 choose 3 subsets! Note: They all have a “5.” Add “5” to every subset. show

The 4 choose 3 subsets are subsets of size 3 from the pool {1, 2, 3, 4}. The 4 choose 3 subsets are: {1,2,3} {1,2,4} {1,3,4} {2,3,4} This gives us more of the 5 choose 3 subsets! Note: None have a “5.” Use these as is. show

{1,2,5} {1,3,5} {1,4,5} {2,3,5} {2,4,5} {3,4,5} The 4 choose 2 subsets become: The 4 choose 3 subsets are: {1,2,3} {1,2,4} {1,3,4} {2,3,4} This does constitute all the 5 choose 3 subsets because any subset of size 3 from the pool {1,2,3,4,5} will either be of the first type (have a 5 and two elements from {1,2,3,4}) or of the second type (be made of of three elements from {1,2,3,4,}). show

Now, in general, We will build the subsets from the subsets and subsets

The n-1 choose r-1 subsets are subsets of size r-1 from the pool . To each of these subsets, add the element . The n-1 choose r subsets are subsets of size r from the pool . Use these subsets as is. Putting all these subsets together we get all the n choose r subsets. show

Therefore, Characterization #1 and #2 are equivalent! This establishes: Therefore, Characterization #1 and #2 are equivalent! Show me Characterization #3