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Jan 26, 2023 •561 likes •1.72k views

Four of a Kind For the Love of Math and Computer Science Allen OHara For the Love of Math and Computer Science Four of a Kind Problem 1 Multiplication takes two numbers and produces a third number. For the Love of Math and Computer Science

  1. (ab) ((ab)c) (a(bc)) (((ab)c)d) ((a(bc))d) ((ab)(cd)) (a((bc)d)) (a(b(cd))) ((((ab)c)d)e) (((a(bc))d)e) (((ab)(cd))e) (((ab)c)(de)) ((a((bc)d))e) ((a(b(cd)))e) ((a(bc))(de)) ((ab)((cd)e)) ((ab)(c(de))) (a(((bc)d)e)) (a((b(cd))e)) (a((bc)(de))) (a(b((cd)e))) (a(b(c(de)))) (((((ab)c)d)e)f) ((((a(bc))d)e)f) ((((ab)(cd))e)f) ((((ab)c)(de))f) ((((ab)c)d)(ef)) (((a((bc)d))e)f) (((a(b(cd)))e)f) (((a(bc))(de))f) (((a(bc))d)(ef)) (((ab)((cd)e))f) (((ab)(c(de)))f) (((ab)(cd))(ef)) (((ab)c)((de)f)) (((ab)c)(d(ef))) ((a(((bc)d)e))f) ((a((b(cd))e))f) ((a((bc)(de)))f) ((a((bc)d))(ef)) ((a(b((cd)e)))f) ((a(b(c(de))))f) ((a(b(cd)))(ef)) ((a(bc))((de)f)) ((a(bc))(d(ef))) ((ab)(((cd)e)f)) ((ab)((c(de))f)) ((ab)((cd)(ef))) ((ab)(c((de)f))) ((ab)(c(d(ef)))) (a((((bc)d)e)f)) (a(((b(cd))e)f)) (a(((bc)(de))f)) (a(((bc)d)(ef))) (a((b((cd)e))f)) (a((b(c(de)))f)) (a((b(cd))(ef))) (a((bc)((de)f))) (a((bc)(d(ef)))) (a(b(((cd)e)f))) (a(b((c(de))f))) (a(b((cd)(ef)))) (a(b(c((de)f)))) (a(b(c(d(ef))))) For the Love of Math and Computer Science Four of a Kind

  2. Problem 2 Imagine a staircase consisting of n equal steps. For the Love of Math and Computer Science Four of a Kind

  3. Problem 2 Imagine a staircase consisting of n equal steps. For the Love of Math and Computer Science Four of a Kind

  4. Problem 2 Imagine a staircase consisting of n equal steps. How many ways are there to tile the staircase with exactly n rectangles? For the Love of Math and Computer Science Four of a Kind

  5. For the Love of Math and Computer Science Four of a Kind

  6. For the Love of Math and Computer Science Four of a Kind

  7. For the Love of Math and Computer Science Four of a Kind

  8. For the Love of Math and Computer Science Four of a Kind

  9. Problem 3 Consider a regular n -gon. For the Love of Math and Computer Science Four of a Kind

  10. Problem 3 Consider a regular n -gon. For the Love of Math and Computer Science Four of a Kind

  11. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles.

  12. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles.

  13. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles.

  14. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles.

  15. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles.

  16. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles. For the Love of Math and Computer Science Four of a Kind

  17. Problem 3 By connecting vertices so that the lines inside the polygon don’t cross, we can break the n -gon into triangles. The end result is called a triangulation. For the Love of Math and Computer Science Four of a Kind

  18. Problem 3 For a given regular n -gon, how many triangulations are there, assuming that rotations and reflections are distinct? For the Love of Math and Computer Science Four of a Kind

  19. For the Love of Math and Computer Science Four of a Kind

  20. For the Love of Math and Computer Science Four of a Kind

  21. For the Love of Math and Computer Science Four of a Kind

  22. For the Love of Math and Computer Science Four of a Kind

  23. Problem 4 A binary tree is a rooted tree where each node has exactly 0, 1, or 2 children. For the Love of Math and Computer Science Four of a Kind

  24. Problem 4 A binary tree is a rooted tree where each node has exactly 0, 1, or 2 children. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● For the Love of Math and Computer Science Four of a Kind

  25. Problem 4 A binary tree is a rooted tree where each node has exactly 0, 1, or 2 children. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● A node with no children is called a leaf. For the Love of Math and Computer Science Four of a Kind

  26. Problem 4 A full binary tree is one where every node has either 0 or 2 children. For the Love of Math and Computer Science Four of a Kind

  27. Problem 4 A full binary tree is one where every node has either 0 or 2 children. How many full binary trees are there with n leaves? For the Love of Math and Computer Science Four of a Kind

  28. For the Love of Math and Computer Science Four of a Kind

  29. For the Love of Math and Computer Science Four of a Kind

  30. For the Love of Math and Computer Science Four of a Kind

  31. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● For the Love of Math and Computer Science Four of a Kind

  32. We may notice that the same numbers keep showing up. For the Love of Math and Computer Science Four of a Kind

  33. We may notice that the same numbers keep showing up. 1, 2, 5, 14, 42, For the Love of Math and Computer Science Four of a Kind

  34. We may notice that the same numbers keep showing up. 1, 2, 5, 14, 42, 132, 429, 1430, ... For the Love of Math and Computer Science Four of a Kind

  35. We may notice that the same numbers keep showing up. 1, 2, 5, 14, 42, 132, 429, 1430, ... Having counted up the individual trees/triangulations/associations/tilings we can see they are the same numbers, but what does that tell us really? For the Love of Math and Computer Science Four of a Kind

  36. ● ● ● ● ● 1 ● ● For the Love of Math and Computer Science Four of a Kind

  37. ● ● ● ● ● 1 ● ● ● ● ● ● ● 2 ● ● For the Love of Math and Computer Science Four of a Kind

  38. ● ● ● ● ● 1 ● ● ● ● ● ● ● 2 ● ● ● ● ● 3 ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ● 5 ● ● For the Love of Math and Computer Science Four of a Kind

  39. ● ● ● ● ● 1 ● ● ● ● ● ● ● 2 ● ● ● ● ● 3 ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● 5 ● ● For the Love of Math and Computer Science Four of a Kind

  40. Is there a way to show a deeper connection? Beyond counting? For the Love of Math and Computer Science Four of a Kind

  41. Is there a way to show a deeper connection? Beyond counting? What we’re looking for is a way to turn a binary tree into an association, or way to turn a staircase tiling into a triangulation. For the Love of Math and Computer Science Four of a Kind

  42. Problem 1 = Problem 4 Can we match associations and binary trees in a meaningful way? ((ab)(cd)) (a((bc)d)) ((a(bc))d) (a(b(cd))) (((ab)c)d) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● For the Love of Math and Computer Science Four of a Kind

  43. For the Love of Math and Computer Science Four of a Kind

  44. From left to right label the leaves of the tree with the variables in the expression. Nodes which are the children of the same parent node get their expressions parenthesized. For the Love of Math and Computer Science Four of a Kind

  45. For the Love of Math and Computer Science Four of a Kind

  46. Problem 4 = Problem 2 If we show these two problems are equivalent we have also shown Problem 1 = Problem 2. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● For the Love of Math and Computer Science Four of a Kind

  47. For the Love of Math and Computer Science Four of a Kind

  48. Rotate the staircase so the lower right corner is in the top middle. Remove the actual steps, revealing a binary tree. For the Love of Math and Computer Science Four of a Kind

  49. For the Love of Math and Computer Science Four of a Kind

  50. Problem 3 = Problem 1 The final connection we need would allow us to conclude Problem 3 = Problem 4 and Problem 3 = Problem 2, as well. (((ab)c)d) ((a(bc))d) ((ab)(cd)) (a((bc)d)) (a(b(cd))) For the Love of Math and Computer Science Four of a Kind

  51. For the Love of Math and Computer Science Four of a Kind

  52. Fix a side of the n -gon. Proceeding clockwise, label the remaining sides with the variables. Label the sides of the triangulation with the parenthesized expressions of the other two sides. For the Love of Math and Computer Science Four of a Kind

  53. For the Love of Math and Computer Science Four of a Kind

  54. Problem 3 is the same as Problem 1 is the same as Problem 4 is the same as Problem 2. For the Love of Math and Computer Science Four of a Kind

  55. Problem 3 is the same as Problem 1 is the same as Problem 4 is the same as Problem 2. All the problems are really the same, just viewed in a different way! For the Love of Math and Computer Science Four of a Kind

  56. Mathematical Illusion What we have is a sort of mathematical illusion . For the Love of Math and Computer Science Four of a Kind

  57. Mathematical Illusion What we have is a sort of mathematical illusion . For the Love of Math and Computer Science Four of a Kind

  58. Mathematical Illusion What we have is a sort of mathematical illusion . The structures change upon perspective, but the underlying connections are all there. For the Love of Math and Computer Science Four of a Kind

  59. Problem 5?? Ian is travelling from the library to the math building across campus. For the Love of Math and Computer Science Four of a Kind

  60. Problem 5?? Ian is travelling from the library to the math building across campus. Campus is laid out in an n × n grid with one-way paths going north and east. For the Love of Math and Computer Science Four of a Kind

  61. Problem 5?? Ian is travelling from the library to the math building across campus. Campus is laid out in an n × n grid with one-way paths going north and east. ● ● For the Love of Math and Computer Science Four of a Kind

  62. Problem 5?? Having recently angered some engineers, Ian must be careful to not walk onto their turf! He must plan his trip so he never goes north of the demarcation line ● ● For the Love of Math and Computer Science Four of a Kind

  63. Problem 5?? Having recently angered some engineers, Ian must be careful to not walk onto their turf! He must plan his trip so he never goes north of the demarcation line ● ● In how many ways can Ian get to the math building? For the Love of Math and Computer Science Four of a Kind

  64. For the Love of Math and Computer Science Four of a Kind

  65. For the Love of Math and Computer Science Four of a Kind

  66. For the Love of Math and Computer Science Four of a Kind

  67. Catalan Numbers This sequence: 1, 2, 5, 14, 42, 132, 429, ... is known as the Catalan numbers. They appear in a large number of counting problems. We’ve already seen some of them, but there are many more. C n = n + 1 ( 2 n 1 n ) For the Love of Math and Computer Science Four of a Kind

  68. More Problems A Dyck word is a string of n Xs and n Ys arranged so that if we read the string from left to right the number of Xs read is always at least as big as the number of Ys read thus far. For the Love of Math and Computer Science Four of a Kind

  69. More Problems A Dyck word is a string of n Xs and n Ys arranged so that if we read the string from left to right the number of Xs read is always at least as big as the number of Ys read thus far. XY For the Love of Math and Computer Science Four of a Kind

  70. More Problems A Dyck word is a string of n Xs and n Ys arranged so that if we read the string from left to right the number of Xs read is always at least as big as the number of Ys read thus far. XY XXY Y , XY XY For the Love of Math and Computer Science Four of a Kind

  71. More Problems A Dyck word is a string of n Xs and n Ys arranged so that if we read the string from left to right the number of Xs read is always at least as big as the number of Ys read thus far. XY XXY Y , XY XY XXXY Y Y , XXY XY Y , XXY Y XY , XY XXY Y , XY XY XY For the Love of Math and Computer Science Four of a Kind

  72. More Problems Imagine 2 chess pawns of the same colour, confined to a single column of a chess board, with n space below them. For the Love of Math and Computer Science Four of a Kind

  73. For the Love of Math and Computer Science Four of a Kind

  74. For the Love of Math and Computer Science Four of a Kind

  75. For the Love of Math and Computer Science Four of a Kind

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