full transcript

## Unscramble the Blue Letters

This may look like a neatly arranged stack of numbers, but it's actually a mathematical treasure trove. Indian mathematicians called it the Staircase of Mount Meru. In Iran, it's the Khayyam Triangle. And in cnhia, it's Yang Hui's Triangle. To much of the wsrteen world, it's known as Pascal's Triangle after French mathematician Blaise paacsl, which seems a bit unfair since he was clearly late to the ptary, but he still had a lot to ctiturobne. So what is it about this that has so intrigued maicmihnaattes the world over? In short, it's full of patterns and secrets. First and foremost, there's the pattern that generates it. Start with one and imagine invisible zeros on either side of it. Add them together in pairs, and you'll generate the next row. Now, do that again and again. Keep going and you'll wind up with something like this, though really Pascal's Triangle goes on infinitely. Now, each row corresponds to what's called the coefficients of a binomial expansion of the form (x+y)^n, where n is the number of the row, and we start counting from zero. So if you make n=2 and expand it, you get (x^2) + 2xy + (y^2). The ceiecifftons, or numbers in fornt of the varaleibs, are the same as the numbers in that row of Pascal's Triangle. You'll see the same thing with n=3, which expands to this. So the triangle is a quick and easy way to look up all of these coefficients. But there's much more. For example, add up the numbers in each row, and you'll get successive powers of two. Or in a given row, treat each number as part of a deiamcl expansion. In other words, row two is (1x1) + (2x10) + (1x100). You get 121, which is 11^2. And take a look at what happens when you do the same thing to row six. It adds up to 1,771,561, which is 11^6, and so on. There are also geometric applications. Look at the dalongais. The first two aren't very interesting: all ones, and then the positive integers, also known as natural numbers. But the numbers in the next diagonal are called the triangular nmeburs because if you take that many dots, you can stack them into equilateral triangles. The next diagonal has the tetrahedral numbers because sliarilmy, you can stack that many spheres into tdeheratra. Or how about this: shade in all of the odd numbers. It doesn't look like much when the triangle's small, but if you add tuohdanss of rows, you get a frctaal known as Sierpinski's tgrnlaie. This triangle isn't just a mtecmaihaatl work of art. It's also quite useful, especially when it comes to probability and caunilaclots in the domain of combinatorics. Say you want to have five children, and would like to know the probability of having your dream family of three girls and two boys. In the binomial expansion, that corresponds to girl plus boy to the fifth power. So we look at the row five, where the first number corresponds to five girls, and the last corresponds to five boys. The third numebr is what we're looking for. Ten out of the sum of all the possibilities in the row. so 10/32, or 31.25%. Or, if you're rolndamy picking a five-player basketball team out of a group of twelve friends, how many possible groups of five are there? In combinatoric tmers, this problem would be phrased as twelve choose five, and could be calculated with this fmluora, or you could just look at the sixth element of row twelve on the triangle and get your answer. The patterns in Pascal's Triangle are a testament to the etlnealgy interwoven fabric of mathematics. And it's still revealing fresh secrets to this day. For example, mathematicians recently discovered a way to expand it to these kdins of polynomials. What might we find next? Well, that's up to you.

## Open Cloze

This may look like a neatly arranged stack of numbers, but it's actually a mathematical treasure trove. Indian mathematicians called it the Staircase of Mount Meru. In Iran, it's the Khayyam Triangle. And in _____, it's Yang Hui's Triangle. To much of the _______ world, it's known as Pascal's Triangle after French mathematician Blaise ______, which seems a bit unfair since he was clearly late to the _____, but he still had a lot to __________. So what is it about this that has so intrigued ______________ the world over? In short, it's full of patterns and secrets. First and foremost, there's the pattern that generates it. Start with one and imagine invisible zeros on either side of it. Add them together in pairs, and you'll generate the next row. Now, do that again and again. Keep going and you'll wind up with something like this, though really Pascal's Triangle goes on infinitely. Now, each row corresponds to what's called the coefficients of a binomial expansion of the form (x+y)^n, where n is the number of the row, and we start counting from zero. So if you make n=2 and expand it, you get (x^2) + 2xy + (y^2). The ____________, or numbers in _____ of the _________, are the same as the numbers in that row of Pascal's Triangle. You'll see the same thing with n=3, which expands to this. So the triangle is a quick and easy way to look up all of these coefficients. But there's much more. For example, add up the numbers in each row, and you'll get successive powers of two. Or in a given row, treat each number as part of a _______ expansion. In other words, row two is (1x1) + (2x10) + (1x100). You get 121, which is 11^2. And take a look at what happens when you do the same thing to row six. It adds up to 1,771,561, which is 11^6, and so on. There are also geometric applications. Look at the _________. The first two aren't very interesting: all ones, and then the positive integers, also known as natural numbers. But the numbers in the next diagonal are called the triangular _______ because if you take that many dots, you can stack them into equilateral triangles. The next diagonal has the tetrahedral numbers because _________, you can stack that many spheres into __________. Or how about this: shade in all of the odd numbers. It doesn't look like much when the triangle's small, but if you add _________ of rows, you get a _______ known as Sierpinski's ________. This triangle isn't just a ____________ work of art. It's also quite useful, especially when it comes to probability and ____________ in the domain of combinatorics. Say you want to have five children, and would like to know the probability of having your dream family of three girls and two boys. In the binomial expansion, that corresponds to girl plus boy to the fifth power. So we look at the row five, where the first number corresponds to five girls, and the last corresponds to five boys. The third ______ is what we're looking for. Ten out of the sum of all the possibilities in the row. so 10/32, or 31.25%. Or, if you're ________ picking a five-player basketball team out of a group of twelve friends, how many possible groups of five are there? In combinatoric _____, this problem would be phrased as twelve choose five, and could be calculated with this _______, or you could just look at the sixth element of row twelve on the triangle and get your answer. The patterns in Pascal's Triangle are a testament to the _________ interwoven fabric of mathematics. And it's still revealing fresh secrets to this day. For example, mathematicians recently discovered a way to expand it to these _____ of polynomials. What might we find next? Well, that's up to you.

## Solution

1. kinds
2. front
3. western
4. tetrahedra
5. terms
6. diagonals
7. number
8. triangle
9. thousands
10. decimal
11. mathematicians
12. numbers
13. pascal
14. coefficients
15. randomly
16. variables
17. elegantly
18. calculations
19. mathematical
20. formula
21. party
22. contribute
23. similarly
24. china
25. fractal

## Original Text

This may look like a neatly arranged stack of numbers, but it's actually a mathematical treasure trove. Indian mathematicians called it the Staircase of Mount Meru. In Iran, it's the Khayyam Triangle. And in China, it's Yang Hui's Triangle. To much of the Western world, it's known as Pascal's Triangle after French mathematician Blaise Pascal, which seems a bit unfair since he was clearly late to the party, but he still had a lot to contribute. So what is it about this that has so intrigued mathematicians the world over? In short, it's full of patterns and secrets. First and foremost, there's the pattern that generates it. Start with one and imagine invisible zeros on either side of it. Add them together in pairs, and you'll generate the next row. Now, do that again and again. Keep going and you'll wind up with something like this, though really Pascal's Triangle goes on infinitely. Now, each row corresponds to what's called the coefficients of a binomial expansion of the form (x+y)^n, where n is the number of the row, and we start counting from zero. So if you make n=2 and expand it, you get (x^2) + 2xy + (y^2). The coefficients, or numbers in front of the variables, are the same as the numbers in that row of Pascal's Triangle. You'll see the same thing with n=3, which expands to this. So the triangle is a quick and easy way to look up all of these coefficients. But there's much more. For example, add up the numbers in each row, and you'll get successive powers of two. Or in a given row, treat each number as part of a decimal expansion. In other words, row two is (1x1) + (2x10) + (1x100). You get 121, which is 11^2. And take a look at what happens when you do the same thing to row six. It adds up to 1,771,561, which is 11^6, and so on. There are also geometric applications. Look at the diagonals. The first two aren't very interesting: all ones, and then the positive integers, also known as natural numbers. But the numbers in the next diagonal are called the triangular numbers because if you take that many dots, you can stack them into equilateral triangles. The next diagonal has the tetrahedral numbers because similarly, you can stack that many spheres into tetrahedra. Or how about this: shade in all of the odd numbers. It doesn't look like much when the triangle's small, but if you add thousands of rows, you get a fractal known as Sierpinski's Triangle. This triangle isn't just a mathematical work of art. It's also quite useful, especially when it comes to probability and calculations in the domain of combinatorics. Say you want to have five children, and would like to know the probability of having your dream family of three girls and two boys. In the binomial expansion, that corresponds to girl plus boy to the fifth power. So we look at the row five, where the first number corresponds to five girls, and the last corresponds to five boys. The third number is what we're looking for. Ten out of the sum of all the possibilities in the row. so 10/32, or 31.25%. Or, if you're randomly picking a five-player basketball team out of a group of twelve friends, how many possible groups of five are there? In combinatoric terms, this problem would be phrased as twelve choose five, and could be calculated with this formula, or you could just look at the sixth element of row twelve on the triangle and get your answer. The patterns in Pascal's Triangle are a testament to the elegantly interwoven fabric of mathematics. And it's still revealing fresh secrets to this day. For example, mathematicians recently discovered a way to expand it to these kinds of polynomials. What might we find next? Well, that's up to you.

## Important Words

4. applications
5. arranged
6. art
8. binomial
9. bit
10. blaise
11. boy
12. boys
13. calculated
14. calculations
15. called
16. children
17. china
18. choose
19. coefficients
20. combinatoric
21. combinatorics
22. contribute
23. corresponds
24. counting
25. day
26. decimal
27. diagonal
28. diagonals
29. discovered
30. domain
31. dots
32. dream
33. easy
34. elegantly
35. element
36. equilateral
37. expand
38. expands
39. expansion
40. fabric
41. family
42. find
43. foremost
44. form
45. formula
46. fractal
47. french
48. fresh
49. friends
50. front
51. full
52. generate
53. generates
54. geometric
55. girl
56. girls
57. group
58. groups
59. imagine
60. indian
61. infinitely
62. integers
63. interwoven
64. intrigued
65. invisible
66. iran
67. khayyam
68. kinds
69. late
70. lot
71. mathematical
72. mathematician
73. mathematicians
74. mathematics
75. meru
76. mount
77. natural
78. neatly
79. number
80. numbers
81. odd
82. pairs
83. part
84. party
85. pascal
86. pattern
87. patterns
88. phrased
89. picking
90. polynomials
91. positive
92. possibilities
93. power
94. powers
95. probability
96. problem
97. quick
98. randomly
99. revealing
100. row
101. rows
102. secrets
104. short
105. side
106. similarly
107. sixth
108. small
109. spheres
110. stack
111. staircase
112. start
113. successive
114. sum
115. team
116. ten
117. terms
118. testament
119. tetrahedra
120. tetrahedral
121. thousands
122. treasure
123. treat
124. triangle
125. triangles
126. triangular
127. trove
128. twelve
129. unfair
130. variables
131. western
132. wind
133. words
134. work
135. world
136. yang
137. zeros