“More Really Cool Things Happening in Pascal’s Triangle – Part III”

1. “More Really Cool Things Happening in Pascal’s Triangle – Part III” Jim Olsen Western Illinois University

2. Outline 0. What kind of session will this be? • Review of some points from the first two talks on Pascal’s Triangle • Look in higher dimensions. • Look at relationships involving triangular numbers. • Including the infinite sum of the reciprocals of the triangular numbers

3. 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. • Your creativity and further discussion will connect this to lesson planning, NCLB, standards, etc.

4. ReviewTriangular numbers

5. 1 +2 +3 +4 +5 +6 +7 +8 +9 Let’s Build the 9th Triangular Number

6. n(n+1) 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. Characterization #5 The Hockey Stick Principle

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

10. Characterization #7 The second diagonal are the triangular numbers. Why? Because we use the Hockey Stick Principle to sum up stick numbers.

11. 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.

12. Characterization #16 The fourth diagonal are the Pentatope numbers. Why? Because we use the Hockey Stick Principle to sum up triangular numbers….er…what

13. Pentatopes • 4 Dimensional • Formed by taking a tetrahedron and adding a point along the fourth dimension through the center of the tetrahedron. The fifth point is equidistant from the original vertices.

14. A Pentatope has • 5 three-dimensional facets, called “cells,” each of which is a tetrahedron • 10 faces • 10 edges • 5 vertices

15. A Pentatope has • 5 three-dimensional facets, called “cells,” each of which is a tetrahedron • 10 faces • 10 edges • 5 vertices Why 10 faces and 10 edges? http://eusebeia.dyndns.org/4d/5-cell.html

16. Next few Pentalope Numbers

17. Characterization #17 The numbers in the nth row of Pascal’s Triangle are the coefficients of This is the binomial theorem (finally), which I won’t prove but let’s think about it using Characterization #9

18. For future study Characterization #18 The middle number in the even rows, divided by the row number plus 1 is a Catalan Number. That is, the Catalan Numbers can be computed using M. Gardner, Mathematical Games, Scientific American, 244 (June 1976), pp. 120-125.

19. 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

20. Solutions to Tower of Hanoi

21. 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:

22. Look at the Sequence as the disks

23. Look at the Sequence as the disks What does it look like?

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

25. Solutions to Tower of HanoiCan you see the ruler markings?

26. Ruler Markings Solution to Tower of Hanoi

27. What is Sierpinski’s Gasket? http://www.shodor.org/interactivate/activities/gasket/ It is a fractal because it is self-similar.

28. by Paul Bourke More Sierpinski Gasket/Triangle Applets and Graphics http://howdyyall.com/Triangles/ShowFrame/ShowGif.cfm http://www.arcytech.org/java/fractals/sierpinski.shtml

29. Characterization #13 If 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

30. Characterization #14 Sierpinski’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.

31. 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.

32. The solution to Tower of Hanoi is given by moving from the top node to the lower right corner.

33. Ruler Markings Solution to Tower of Hanoi Sierpinski Wire Frame

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

35. Characterization #12.1 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. Count in Base-2

36. 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

37. Ruler Markings Solution to Tower of Hanoi Sierpinski Wire Frame Binary Numbers

38. More Information September 1978, Mathematics Teacher, pp.505-510, by J. Staib and L. Staib. “The Pascal Pyramid.” http://www.wiu.edu/users/mfjro1/wiu/tea/pascal-tri.htm

39. Thank You Jim Olsen Western Illinois University jr-olsen@wiu.edu faculty.wiu.edu/JR-Olsen/wiu/