More Really Cool Things Happening in Pascal’s Triangle

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More 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?
• Review of some points from the first talk on
  Pascals Triangle and Counting Toothpicks in the
  Twelve Days of Christmas Tetrahedron
• Two Questions posed.
• Characterizations involving Tower of Hanoi,
  Sierpinski, and _______ and _______.
• A couple more interesting characterizations.
• Two Questions solved.

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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.
• Want to look at some big ideas and make some
  connections.
• I will continually explain things at various
  levels and varying amounts of detail.
• Resources are available, if you want more.
• Your creativity and further discussion will
  connect this to lesson planning, NCLB, standards,
  etc.

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ReviewTriangular numbers
(Review)
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Lets Build the 9th Triangular Number
(Review)
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n(n1) Take half. Each Triangle has n(n1)/2
n
n1
(Review)
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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 Characterizations

• Char. 1 First Definition Get each number in a
  row from the two numbers diagonally above it (and
  begin and end each row with 1). This is the
  standard way to generate Pascals Triangle.

(Review)
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• Char. 2 Second Definition A Table of
  Combinations or Numbers of Subsets
• (Characterization 1 and characterization 2 can
  be shown to be equivalent)
• Char. 3 Symmetry

(Review)
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(Review)
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• Char. 4 The total of row n
• the Total Number of Subsets (from a set of
  size n)
• 2n

(Review)
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• Char. 5 The Hockey Stick Principle

(Review)
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• Char. 6 The first diagonal are the stick
  numbers.
• Char. 7 second diagonal are the triangular
  numbers.

(Review)
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• Char. 8 The third diagonal are the tetrahedral
  numbers.

(Review)
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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?
(Review)
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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!
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• Char.9 This is actually a table of permutations
  (permutations with repetitions).
• Char. 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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• Char. 11 The fourth diagonal lists the number
  of quadrilaterals formed by n points on a circle.

(Review)
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2. Two Questions posed

• What is the sum of the squares of odd numbers (or
  squares of even numbers)?
• What is the difference of the squares of two
  consecutive triangular numbers?

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3. Characterizations involving Tower of Hanoi,
Sierpinski, and _______ and _______.

• Solve Tower of Hanoi.
• What do we know? Brainstorm.
• http//www.mazeworks.com/hanoi/index.htm

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Solutions to Tower of Hanoi
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Characterization 12

• The sum of the first n rows of Pascals 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 Pascals Triangle
  (which are rows 0 to n-1) is one less than the
  sum of the nth row. (by Char.4)
• Equivalently

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Look at the Sequence as the disks
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Look at the Sequence as the disks
What does it look like?
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Look at the Sequence as the disks
A ruler!
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Solutions to Tower of HanoiCan you see the ruler
markings?
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(No Transcript)
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What is Sierpinskis Gasket?

• http//www.shodor.org/interactivate/activities/gas
  ket/

It is a fractal because it is self-similar.
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More Sierpinski Gasket/Triangle Applets and
Graphics

• http//howdyyall.com/Triangles/ShowFrame/ShowGif.c
  fm
• http//www.arcytech.org/java/fractals/sierpinski.s
  html

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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/paco
ima.html
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Characterization 13

• If you color the odd numbers red and the even
  numbers black in Pascals Triangle, you get a
  (red) Sierpinski Gasket.

http//www.cecm.sfu.ca/cgi-bin/organics/pascalform
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Characterization 14

• Sierpinskis Gasket, with 2n rows, provides a
  solution (and the best solution) to the Tower of
  Hanoi problem with n disks.

At each (red) colored node in Sierpinskis Gasket
assign an n-tuple of 1s, 2s, and 3s (numbers
stand for the pin/tower number). The first number
in the n-tuple tells where the a-disk goes (the
smallest disk). The second number in the n-tuple
tells where the b-disk goes (the second
disk). Etc.
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Maybe we should call it Sierpinskis Wire Frame
The solution to Tower of Hanoi is given by moving
from the top node to the lower right corner.
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The solution to Tower of Hanoi is given by moving
from the top node to the lower right corner.
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(No Transcript)
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But isnt all of this

• Yes/No..On/off
• Binary
• Base Two

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Characterization 12.1

• The sum of the first n rows of Pascals Triangle
  (which are rows 0 to n-1) is the number of
  non-zero base-2 numbers with n digits.

Count in Base-2
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• 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?
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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

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Binary Numbers
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4. A Couple More Interesting Characterizations.
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Characterization 15

• By adding up numbers on diagonals in Pascals
  Triangle, you get the Fibonacci numbers.

This works because of Characterization 1 (and
the fact that rows begin and end with 1).
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Characterization 16

• To get the numbers in any row (row n), start with
  1 and successively multiply by

For example, to generate row 6.
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5. Two Questions Answered

• What is the sum of the squares of odd numbers (or
  squares of even numbers)?

See Model.
Answer A tetrahedron. In fact, or
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Two Questions Answered (cont.)

• What is the difference of the squares of two
  consecutive triangular numbers?

See Model.
Answer A cube. In fact, n3.
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More Information

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

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• Thank you.

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