SLIDE 1 Four of a Kind
For the Love of Math and Computer Science Allen O’Hara
For the Love of Math and Computer Science Four of a Kind
SLIDE 2 Problem 1
Multiplication takes two numbers and produces a third number.
For the Love of Math and Computer Science Four of a Kind
SLIDE 3 Problem 1
Multiplication takes two numbers and produces a third number. 2 × 3 = 6
For the Love of Math and Computer Science Four of a Kind
SLIDE 4 Problem 1
When we do repeated multiplications, we have choices as to what pairs of numbers we multiply first. 2 × 3 × 5 2 × 3 × 5
For the Love of Math and Computer Science Four of a Kind
SLIDE 5 Problem 1
When we do repeated multiplications, we have choices as to what pairs of numbers we multiply first. 2 × 3 × 5 2 × 3 × 5 = (2 × 3) × 5 = 2 × (3 × 5)
For the Love of Math and Computer Science Four of a Kind
SLIDE 6 Problem 1
When we do repeated multiplications, we have choices as to what pairs of numbers we multiply first. 2 × 3 × 5 2 × 3 × 5 = (2 × 3) × 5 = 2 × (3 × 5) = 6 × 5 = 2 × 15
For the Love of Math and Computer Science Four of a Kind
SLIDE 7 Problem 1
When we do repeated multiplications, we have choices as to what pairs of numbers we multiply first. 2 × 3 × 5 2 × 3 × 5 = (2 × 3) × 5 = 2 × (3 × 5) = 6 × 5 = 2 × 15 = 30 = 30
For the Love of Math and Computer Science Four of a Kind
SLIDE 8 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17
For the Love of Math and Computer Science Four of a Kind
SLIDE 9 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17
For the Love of Math and Computer Science Four of a Kind
SLIDE 10 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17
For the Love of Math and Computer Science Four of a Kind
SLIDE 11 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17)
For the Love of Math and Computer Science Four of a Kind
SLIDE 12 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 13 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 14 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 15 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221 = 6 × (385 × 13) × 17 = (6 × 35) × 11 × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 16 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221 = 6 × (385 × 13) × 17 = (6 × 35) × 11 × 221 = 6 × 5005 × 17 = 210 × 11 × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 17 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221 = 6 × (385 × 13) × 17 = (6 × 35) × 11 × 221 = 6 × 5005 × 17 = 210 × 11 × 221 = 6 × (5005 × 17) = (210 × 11) × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 18 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221 = 6 × (385 × 13) × 17 = (6 × 35) × 11 × 221 = 6 × 5005 × 17 = 210 × 11 × 221 = 6 × (5005 × 17) = (210 × 11) × 221 = 6 × 85085 = 2310 × 221
For the Love of Math and Computer Science Four of a Kind
SLIDE 19 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221 = 6 × (385 × 13) × 17 = (6 × 35) × 11 × 221 = 6 × 5005 × 17 = 210 × 11 × 221 = 6 × (5005 × 17) = (210 × 11) × 221 = 6 × 85085 = 2310 × 221 = (6 × 85085) = (2310 × 221)
For the Love of Math and Computer Science Four of a Kind
SLIDE 20 Problem 1
2 × 3 × 5 × 7 × 11 × 13 × 17 2 × 3 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (5 × 7) × 11 × 13 × 17 = (2 × 3) × 5 × 7 × 11 × 13 × 17 = 2 × 3 × 35 × 11 × 13 × 17 = 6 × 5 × 7 × 11 × 13 × 17 = 2 × 3 × (35 × 11) × 13 × 17 = 6 × 5 × 7 × 11 × (13 × 17) = 2 × 3 × 385 × 13 × 17 = 6 × 5 × 7 × 11 × 221 = (2 × 3) × 385 × 13 × 17 = 6 × (5 × 7) × 11 × 221 = 6 × 385 × 13 × 17 = 6 × 35 × 11 × 221 = 6 × (385 × 13) × 17 = (6 × 35) × 11 × 221 = 6 × 5005 × 17 = 210 × 11 × 221 = 6 × (5005 × 17) = (210 × 11) × 221 = 6 × 85085 = 2310 × 221 = (6 × 85085) = (2310 × 221) = 510510 = 510510
For the Love of Math and Computer Science Four of a Kind
SLIDE 21 Problem 1
Our choices lead to different ways to bracket (associate) the expression. ((2 × 3) × ((((5 × 7) × 11) × 13) × 17)) ((((2 × 3) × (5 × 7)) × 11) × (13 × 17))
For the Love of Math and Computer Science Four of a Kind
SLIDE 22 Problem 1
Our choices lead to different ways to bracket (associate) the expression. ((2 × 3) × ((((5 × 7) × 11) × 13) × 17)) ((((2 × 3) × (5 × 7)) × 11) × (13 × 17)) Suppose we have n numbers in our product. How many ways can the expression be bracketed, without reordering the factors?
For the Love of Math and Computer Science Four of a Kind
SLIDE 23 For the Love of Math and Computer Science Four of a Kind
SLIDE 24 For the Love of Math and Computer Science Four of a Kind
SLIDE 25 For the Love of Math and Computer Science Four of a Kind
SLIDE 26 (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
SLIDE 27 Problem 2
Imagine a staircase consisting of n equal steps.
For the Love of Math and Computer Science Four of a Kind
SLIDE 28 Problem 2
Imagine a staircase consisting of n equal steps.
For the Love of Math and Computer Science Four of a Kind
SLIDE 29 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
SLIDE 30 For the Love of Math and Computer Science Four of a Kind
SLIDE 31 For the Love of Math and Computer Science Four of a Kind
SLIDE 32 For the Love of Math and Computer Science Four of a Kind
SLIDE 33 For the Love of Math and Computer Science Four of a Kind
SLIDE 34 Problem 3
Consider a regular n-gon.
For the Love of Math and Computer Science Four of a Kind
SLIDE 35 Problem 3
Consider a regular n-gon.
For the Love of Math and Computer Science Four of a Kind
SLIDE 36
Problem 3
By connecting vertices so that the lines inside the polygon don’t cross, we can break the n-gon into triangles.
SLIDE 37
Problem 3
By connecting vertices so that the lines inside the polygon don’t cross, we can break the n-gon into triangles.
SLIDE 38
Problem 3
By connecting vertices so that the lines inside the polygon don’t cross, we can break the n-gon into triangles.
SLIDE 39
Problem 3
By connecting vertices so that the lines inside the polygon don’t cross, we can break the n-gon into triangles.
SLIDE 40
Problem 3
By connecting vertices so that the lines inside the polygon don’t cross, we can break the n-gon into triangles.
SLIDE 41 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
SLIDE 42 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
SLIDE 43 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
SLIDE 44 For the Love of Math and Computer Science Four of a Kind
SLIDE 45 For the Love of Math and Computer Science Four of a Kind
SLIDE 46 For the Love of Math and Computer Science Four of a Kind
SLIDE 47 For the Love of Math and Computer Science Four of a Kind
SLIDE 48 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
SLIDE 49 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
SLIDE 50 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
SLIDE 51 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
SLIDE 52 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
SLIDE 53 For the Love of Math and Computer Science Four of a Kind
SLIDE 54 For the Love of Math and Computer Science Four of a Kind
SLIDE 55 For the Love of Math and Computer Science Four of a Kind
SLIDE 56
For the Love of Math and Computer Science Four of a Kind
SLIDE 57 We may notice that the same numbers keep showing up.
For the Love of Math and Computer Science Four of a Kind
SLIDE 58 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
SLIDE 59 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
SLIDE 60 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
SLIDE 61
For the Love of Math and Computer Science Four of a Kind
SLIDE 62
For the Love of Math and Computer Science Four of a Kind
SLIDE 63
For the Love of Math and Computer Science Four of a Kind
SLIDE 64
For the Love of Math and Computer Science Four of a Kind
SLIDE 65 Is there a way to show a deeper connection? Beyond counting?
For the Love of Math and Computer Science Four of a Kind
SLIDE 66 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
SLIDE 67 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
SLIDE 68 For the Love of Math and Computer Science Four of a Kind
SLIDE 69 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
SLIDE 70 For the Love of Math and Computer Science Four of a Kind
SLIDE 71 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
SLIDE 72 For the Love of Math and Computer Science Four of a Kind
SLIDE 73 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
SLIDE 74 For the Love of Math and Computer Science Four of a Kind
SLIDE 75 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
SLIDE 76 For the Love of Math and Computer Science Four of a Kind
SLIDE 77 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
SLIDE 78 For the Love of Math and Computer Science Four of a Kind
SLIDE 79 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
SLIDE 80 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
SLIDE 81 Mathematical Illusion
What we have is a sort of mathematical illusion.
For the Love of Math and Computer Science Four of a Kind
SLIDE 82 Mathematical Illusion
What we have is a sort of mathematical illusion.
For the Love of Math and Computer Science Four of a Kind
SLIDE 83 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
SLIDE 84 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
SLIDE 85 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
SLIDE 86 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
SLIDE 87 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
SLIDE 88 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
SLIDE 89 For the Love of Math and Computer Science Four of a Kind
SLIDE 90 For the Love of Math and Computer Science Four of a Kind
SLIDE 91 For the Love of Math and Computer Science Four of a Kind
SLIDE 92 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. Cn =
1 n+1 (2n n )
For the Love of Math and Computer Science Four of a Kind
SLIDE 93 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
SLIDE 94 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
SLIDE 95 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
SLIDE 96 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
SLIDE 97 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
SLIDE 98 For the Love of Math and Computer Science Four of a Kind
SLIDE 99 For the Love of Math and Computer Science Four of a Kind
SLIDE 100 For the Love of Math and Computer Science Four of a Kind
SLIDE 101 For the Love of Math and Computer Science Four of a Kind
SLIDE 102 For the Love of Math and Computer Science Four of a Kind
SLIDE 103 For the Love of Math and Computer Science Four of a Kind
SLIDE 104 More Problems
Imagine 2 chess pawns of the same colour, confined to a single column of a chess board, with n space infront of them. How many lists of valid chess instructions can be made that get both pawns to the bottom of the column.
For the Love of Math and Computer Science Four of a Kind
SLIDE 105 More Problems
Picture a 2 × n grid.
For the Love of Math and Computer Science Four of a Kind
SLIDE 106 More Problems
Picture a 2 × n grid. In how many ways can the numbers from 1 to 2n be placed in the grid so that every row and column is read in increasing order?
For the Love of Math and Computer Science Four of a Kind
SLIDE 107 1 2 1 2 3 4 1 3 2 4 1 2 3 4 5 6 1 2 4 3 5 6 1 2 5 3 4 6 1 3 4 2 5 6 1 3 5 2 4 6
For the Love of Math and Computer Science Four of a Kind
SLIDE 108 More Problems
Imagine the vertices of a regular n-gon.
- For the Love of Math and Computer Science
Four of a Kind
SLIDE 109 More Problems
Imagine the vertices of a regular n-gon.
- How many ways are there to group the vertices of the n-gon so that the groups do
not overlap?
For the Love of Math and Computer Science Four of a Kind
SLIDE 110 More Problems
For the Love of Math and Computer Science Four of a Kind
SLIDE 111 Bell Numbers
For the Love of Math and Computer Science Four of a Kind
SLIDE 112 Bell Numbers
Imagine the vertices of a regular n-gon. How many partitions (possibly
- verlapping) of the vertices are there if rotations and reflections are considered
distinct? If a positive, squarefree number N has n prime factors, in how many different ways can N be written as a product of factors greater than 1? Determine the number of rhyming schemes to an n line poem.
For the Love of Math and Computer Science Four of a Kind
SLIDE 113 Fibonacci Numbers
For the Love of Math and Computer Science Four of a Kind
SLIDE 114 Fibonacci Numbers
Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs are there after n months? If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after n years, assuming none die? A male bee has only a mother, whereas a female bee has both a father and a
- mother. Determine the number of bees in each ancestral generation in the family
tree of a male bee.
For the Love of Math and Computer Science Four of a Kind
