Some Really Cool Things Happening in Pascal’s Triangle

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Some Really Cool Things Happening in Pascals
Triangle

• Jim Olsen
• Western Illinois University

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Outline

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

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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
• Im going to move quickly.
• I will continually explain things at various
  levels.

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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. Id like to explore more
  answers to this question.

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Answering non-Math Questions in advance

• Q Isnt 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.

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What kind of session will this be?

• I want to look at the beauty of Pascals Triangle
  and improve understanding by making connections
  using various representations.

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2. Triangular numbers
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In General, there are Polygonal Numbers Or
Figurate Numbers
Example The pentagonal numbers are 1, 5, 12,
22,
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Lets Build the 9th Triangular Number
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Interesting facts about Triangular Numbers

• The Triangular Numbers are the Handshake Numbers

• Which are the number of sides and diagonals of an
  n-gon.

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Number of Handshakes Number of sides and
diagonals of an n-gon.
A
B
E
D
C
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Why are the handshake numbers Triangular?
Lets 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
Its a Triangle !
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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.
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9x10 90 Take half. Each Triangle has 45.
9
9110
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n(n1) Take half. Each Triangle has n(n1)/2
n
n1
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Another Cool Thing about Triangular Numbers

• Put any triangular number together with the next
  bigger (or next smaller).

And you get a Square!
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Eleven Cool Things About Pascals Triangle
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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).

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Example To get the 5th element in row 7, you
add the 4th and 5th element in row 6.
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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?

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etc.
etc.
Five Choose Two
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A, B
A, B
A, C
A, C
A, D
A, D
etc.
etc.
A, B, C, D, E
Form subsets of size Two ? Five Choose Two
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• Therefore, the number of combinations of a
  certain size is the same as the number of subsets
  of that size.

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Characterization 1 and characterization 2 are
equivalent, because
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Characterization 3

• Symmetry or
• Now you have it,
• now you dont.

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Characterization 4

• The total of row n
• the Total Number of Subsets (from a set of
  size n)
• 2n

Why?
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Characterization 5

• The Hockey Stick Principle

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• The Hockey Stick Principle

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Characterization 6

• The first diagonal are the stick numbers.

boring, but a lead-in to
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Characterization 7

• The second diagonal are the triangular numbers.

Why?
Because we use the Hockey Stick Principle to sum
up stick numbers.
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• Now lets add up triangular numbers (use the
  hockey stick principle).
• And we get, the 12 Days of Christmas.
• A Tetrahedron.

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Characterization 8

• The third diagonal are the tetrahedral numbers.

Why?
Because we use the Hockey Stick Principle to sum
up triangular numbers.
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Tetrahedral Numbers are Cool Like Triangular
Numbers

• Do the same things.
• Find a general formula.
• Add up consecutive Tetrahedral Numbers.

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Find a general formula.

• Use Six copies of the tetrahedron !

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Combine Two Consecutive Tetrahedrals

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

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Characterization 9

• This is actually a table of permutations.
• Permutations with repetitions. Two types of
  objects that need to be arranged.
• For Example, lets say we have 2 Red tiles and 3
  Blue tiles and we want to arrange all 5 tiles.
  How many permutations (arrangements) are there?

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For Example, lets 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?(

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Characterization 10
Imagine a pin at each location in the first n
rows of Pascals 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 gtgt 1 7 21 35 35 21 7 1

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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).
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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.
Pascals Triangle gives it to you for any size
grid!
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Characterization 11

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

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• 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.
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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.

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Now to Solve Two Classic Problems

1. If you connect n random points on a circle, how
   many regions do you get? (What is the most
   number of regions?)
2. If you cut a pizza with n random cuts, how many
   regions do you get? (What is the most number of
   regions?)

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

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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)
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If you connect n random points on a circle, how
many regions do you get?
Answer to
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If you connect n random points on a circle, how
many regions do you get?
Answer to
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2. If you cut a pizza with n random cuts, how
many regions do you get?
Answer to
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If you cut a pizza with n random cuts, how many
regions do you get?
Answer to
Example 6 cuts in a pizza give a maximum of 22
pieces.
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A Neat Method to Find Any Figurate Number

• Number example
• Lets find the 6th pentagonal number.

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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
12030 51
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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)
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• Resources,
• Thank you.

• Jim Olsen
• Western Illinois University
• jr-olsen_at_wiu.edu
• faculty.wiu.edu/JR-Olsen/wiu/

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Addendum
First we will show this for a number example n
5 r 3.
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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
1,2,5
1,3,5
1,4,5
2,3,5
2,4,5
3,4,5
5
5
5
5
5
5
This gives us some of the 5 choose 3 subsets!
Note They all have a 5.
Add 5 to every subset.
show
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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
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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,).
This does constitute all the 5 choose 3 subsets
show
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Now, in general,
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Use these subsets as is.
Putting all these subsets together we get all
the n choose r subsets.
show
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Therefore, Characterization 1 and 2 are
equivalent!