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“More Really Cool Things Happening in Pascal’s Triangle”. Jim Olsen Western Illinois University. Outline. 0.What kind of session will this be? Review of some points from the first talk on Pascal’s Triangle and Counting Toothpicks in the Twelve Days of Christmas Tetrahedron - PowerPoint PPT Presentation
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“More Really Cool Things Happening in Pascal’s
Triangle”
Jim Olsen
Western Illinois University
Outline
0. What kind of session will this be?
1. Review of some points from the first talk on Pascal’s Triangle and Counting Toothpicks in the Twelve Days of Christmas Tetrahedron
2. Two Questions posed.
3. Characterizations involving Tower of Hanoi, Sierpinski, and _______ and _______.
4. A couple more interesting characterizations.
5. Two Questions solved.
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.
1. ReviewTriangular numbers
15 10 6 3 1 54321 TTTTT
(Review)
+31+2 +9+8+7+6+5+4
Let’s Build the 9th
Triangular Number
(Review)
459 T
n
n+1
n(n+1)
Take half.
Each Triangle
has n(n+1)/2
2
)1(
nnTn
(Review)
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
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 Pascal’s Triangle.
(Review)
• 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)
7
9
2
9
rn
n
r
n
subsets 10
102
5 2 choose 5
subsets 120
1207
10 7 choose 10
(Review)
• Char. #4: The total of row n
= the Total Number of Subsets (from a set of size n)
= 2n
32215101051 5
n
n
nnnn2...
210
(Review)
• Char. #5: The Hockey Stick Principle
(Review)
• Char. #6: The first diagonal are the “stick” numbers.
• Char. #7: second diagonal are the triangular numbers.
(Review)
• Char. #8: The third diagonal are the tetrahedral numbers.
(Review)
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)
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….
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
• 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 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.
Char. #11: The fourth diagonal lists the number of quadrilaterals formed by n points on a circle.
4
n
(Review)
2. Two Questions posed
1. What is the sum of the squares of odd numbers (or squares of even numbers)?
2. What is the difference of the squares of two consecutive triangular numbers?
?)12(...531 2222 n
?21
2 nn TT
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
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/cgi-bin/organics/pascalform
Characterization #14Sierpinski’s 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 Sierpinski’s Gasket assign an n-tuple of 1’s, 2’s, and 3’s (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.
Maybe we should call it Sierpinski’s Wire Frame
The solution to Tower of Hanoi is given by moving from the top node to the lower right corner.
The solution to Tower of Hanoi is given by moving from the top node to the lower right corner.
Solution to Tower of Hanoi
Sierpinski 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 Wire Frame
1 10 11
100101110111
Binary Numbers
Ruler Markings
4. A Couple More Interesting Characterizations.
Characterization #15By adding up numbers on “diagonals” in Pascal’s
Triangle, you get the Fibonacci numbers.
This works because of
Characterization #1 (and the fact that rows begin and end with 1).
Characterization #16
To get the numbers in any row (row n), start with 1 and successively multiply by
n
nnnn 1,...,
4
3,
3
2,
2
1,
1
For example, to generate row 6.
1
1 6 15 20 15 6 11
6
2
5
3
4
4
3
5
2
6
1
5. Two Questions Answered1. What is the sum of the squares of odd
numbers (or squares of even numbers)?
?)12(...531 2222 n
Answer: A tetrahedron. In fact, or
?)2(...642 2222 n
See Model.
12 nTe 2nTe
Two Questions Answered (cont.)
2. What is the difference of the squares of two consecutive triangular numbers?
?21
2 nn TT
Answer: A cube. In fact, n3.
See Model.
More Information
http://www.wiu.edu/users/mfjro1/wiu/tea/pascal-tri.htm
Jim Olsen
Western Illinois University
faculty.wiu.edu/JR-Olsen/wiu/
•Thank you.