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

2. 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

3. 1. Triangular numbers

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

5. Let’s Build the 9th Triangular Number1
Let’s Build the 9th Triangular Number +2 +3 +4 +5 +6 +7 +8 +9

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

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

8. 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).

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

10. 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?

11. etc.
etc.
Five Choose Two

12. {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

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

15. Characterization #1 and characterization #2 are equivalent, because

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

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

18. Characterization #5
The Hockey Stick Principle

19. The Hockey Stick Principle

20. Characterization #6 The first diagonal are the “stick” numbers.
…boring, but a lead-in to…

21. Characterization #7 The second diagonal are the triangular numbers.
Why?
Because of summing up stick numbers and the Hockey Stick Principle

22. 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

23. Decompose a hexagonal number into 4 triangular numbers.

25. A Neat Method to Find Any Figurate Number
Number example:
Let’s find the 6th pentagonal number.

26. 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
= 51

27. 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)

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

29. 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.

30. Formula for the nth tetrahedral number …see it…

31. From
Proofs
Without
Words

32. Characterization #9
Pascal’s triangle is actually a table of permutations.
Permutations with repetitions. Two types of objects that need to be arranged.

33. 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?…(

34. 2. Characterizations involving Tower of Hanoi, Sierpinski, and _______ and _______.
Solve Tower of Hanoi.
What do we know? Brainstorm.

35. 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

36. 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:

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

38. Look at the Sequence as the disks
Moves Needed
Sequence 3 7
aba c aba
What does it look like?

39. Look at the Sequence as the disks
A ruler!

40. 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

41. Solution to Tower of Hanoi
Ruler Markings
Solution to Tower of Hanoi

42. What is Sierpinski’s Gasket?
It is a fractal because it is self-similar.

43. More Sierpinski Gasket/Triangle Applets and Graphics
by Paul Bourke

44. Vladimir Litt's, seventh grade pre-algebra class from Pacoima Middle School Pacoima, California created the most amazing Sierpinski Triangle.

45. Characterization #13
If you color the odd numbers red and the even numbers black in Pascal’s Triangle, you get a (red) Sierpinski Gasket.

46. Solution to Tower of Hanoi Sierpinski Gasket/Wire Frame
Ruler Markings
Solution to Tower of Hanoi
Sierpinski Gasket/Wire Frame

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

48. 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

49. 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?

50. 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

51. 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

52. 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, , , , …
(“numerous” is an understatement)

53. 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?

54. Number of Toothpicks from each
Category of Nodes
Number of Nodes
Number of Toothpicks from each
Product
Corners
Edges x
Faces xT
Interior Te
Total: 3432
But….
This double counts, so there are 1716 toothpicks!

55. Western Illinois University
Thank You
Jim Olsen
Western Illinois University