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The Art of Counting

The Art of Counting. David M. Bressoud Macalester College St. Paul, MN BAMA, April 12, 2006. This Power Point presentation can be downloaded from www.macalester.edu/~bressoud/talks. Review of binomial coefficients & Pascal’s triangle Slicing cheese

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The Art of Counting

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  1. TheArtofCounting David M. Bressoud Macalester College St. Paul, MN BAMA, April 12, 2006 This Power Point presentation can be downloaded from www.macalester.edu/~bressoud/talks

  2. Review of binomial coefficients & Pascal’s triangle Slicing cheese A problem inspired by Charles Dodgson (aka Lewis Carroll)

  3. Recursion Building the next value from the previous values

  4. Given 5 objects Choose 2 of them How many ways can this be done?

  5. Given 5 objects Choose 2 of them How many ways can this be done? ABCDE AB, AC, AD, AE BC, BD, BE, CD, CE, DE

  6. Given 5 objects Choose 2 of them How many ways can this be done? ABCDE AB, AC, AD, AE BC, BD, BE, CD, CE, DE

  7. +

  8. + + + + + + + + + +

  9. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 “Pascal’s” triangle Published 1654

  10. “Pascal’s” triangle from Siyuan yujian by Zhu Shihjie, 1303 CE Dates to Jia Xian circa 1100 CE, possibly earlier in Baghdad-Cairo or in India.

  11. How many regions do we get if we cut space by 6 planes? George Pólya (1887–1985) Let Us Teach Guessing Math Assoc of America, 1965

  12. How many regions do we get if we cut space by 6 planes? 0 planes: 1 region 1 plane: 2 regions 2 planes: 4 regions 3 planes: 8 regions

  13. How many regions do we get if we cut space by 6 planes? 0 planes: 1 region 1 plane: 2 regions 2 planes: 4 regions 3 planes: 8 regions 4 planes: 15 regions

  14. Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7

  15. Cut a plane by lines Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4

  16. 2 3 7 1 6 4 5

  17. Cut a plane by lines Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 7

  18. Cut a plane by lines Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 7

  19. Cut a plane by lines Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 7 11

  20. Cut a plane by lines Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 7 11 16 22

  21. Cut a plane by lines Cut space by planes Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 8 1 2 4 7 11 16 22

  22. 4th plane cuts each of the previous 3 planes on a line

  23. Cut a plane by lines Cut space by planes Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 8 15 1 2 4 7 11 16 22

  24. 5th plane cuts each of the previous 4 planes on a line

  25. Cut a plane by lines Cut space by planes Cut a line by points 0: 1 1: 2 2: 3 3: 4 4: 5 5: 6 6: 7 1 2 4 8 15 26 ?? 1 2 4 7 11 16 22

  26. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

  27. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

  28. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

  29. line by points plane by lines space by planes 1 1 1 1 2 2 2 1 1 3 4 4 1 2 1 4 7 8 1 3 3 1 5 11 15 1 4 6 4 1 6 16 26 1 5 10 10 5 1 7 22 42 1 6 15 20 15 6 1

  30. Number of regions created when space is cut by k planes:

  31. Number of regions created when space is cut by k planes: Can we make sense of this formula? What formula gives us the number of finite regions? What happens in higher dimensional space and what does that mean?

  32. Charles L. Dodgson aka Lewis Carroll “Condensation of Determinants,” Proceedings of the Royal Society, London 1866

  33. Bill Mills Institute for Defense Analysis Howard Rumsey Dave Robbins

  34. Alternating Sign Matrix: • Every row sums to 1 • Every column sums to 1 • Non-zero entries alternate in sign

  35. Alternating Sign Matrix: • Every row sums to 1 • Every column sums to 1 • Non-zero entries alternate in sign A5 = 429

  36. Monotone Triangle

  37. Monotone Triangle

  38. 1 2 3 4 5 12 13 14 15 23 24 25 34 35 45 123 124 125 134 135 145 234 235 345 1234 1235 1245 1345 2345 12345

  39. 1 2 3 4 5 12 13 14 15 23 24 25 34 35 45 123 124 125 134 135 145 234 235 345 3 1234 1235 1245 1345 2345 12345

  40. 1 2 3 4 5 12 13 14 15 23 24 25 34 35 45 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

  41. 1 2 3 4 5 12 13 14 15 23 24 25 34 35 45 14 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

  42. 1 2 3 4 5 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

  43. 1 2 3 4 5 105 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

  44. 1 2 3 4 5 42 105 135 105 42 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

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