Faculty of Mathematics Centre for Education in

Waterloo, Ontario N2L 3G1 Mathematics and Computing

Grade 6 Math CirclesOctober 18 & 19, 2016

Pascal’s Triangle

Warm Up

Before we begin, let’s do a few exercises!

1. Rewrite the following expressions using exponents:

a) 2× 2× 2× 2× 2

b) 5× 5× 5

c) 12× 12× 12× 12× 12× 12× 12

2. Solve the following and state whether it is a square, cube, or neither:

a) 32

b) 5× 5× 5

c) 20

Who is Pascal?

Blaise Pascal (1623-1662) was a French mathematician with

many published works in different topics of mathematics. He

is best known for his work in 1653 on Traite du triangle

arithmetique which is more famously known as Pascal’s Tri-

angle.

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Pascal’s Triangle

row 0 =⇒ 1

row 1 =⇒ 1 1

row 2 =⇒ 1 2 1

row 3 =⇒ 1 3 3 1

row 4 =⇒ 1 4 6 4 1

row 5 =⇒ 1 5 10 10 5 1

row 6 =⇒row 7 =⇒

To draw Pascal’s triangle, start with 1. In the next row, we have 1, 1. Then in the next row,

1, 2 (⇒ 1 + 1), 1 and so on. Each number within the triangle is the sum of the two numbers

in the row above. Pascal’s Triangle can go on for however long you like!

We call the first row, “row 0” followed by “row 1”, “row 2”, ...

The first number, or term, in each row is called “term 0”, followed by “term 1”, “term 2”,

and so on. For example, let’s take a closer look at row 3.

row 3 =⇒ 1 3 3 1

⇑ ⇑ ⇑ ⇑term 0 term 1 term 2 term 3

Example 1 Find the number given the row number and term number:

a) row 2, term 1 b) row 4, term 2

Example 2 Find the row number and term number of the following:

a) 5 b) 35

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Patterns in Pascal’s Triangle

Diagonals

Check it out. The first diagonal, the term 0 of each row, are all 1s! Not very surprising. The

next diagonal are the counting numbers (1, 2, 3, 4, ...). And the third diagonal contains the

triangular number sequence.

1 ⇐ 1s

1 1 ⇐ Counting Numbers

1 2 1 ⇐ Triangular Numbers

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

The triangular number sequence is made from a pattern of dots that form a triangle. We

can find the next number of the sequence, or the next number of dots for each new triangle,

by adding a row of dots to the previous triangle and counting all the dots.

1 dot 3 dots 6 dots 10 dots

1st triangle 2nd triangle 3rd triangle 4th triangle

Let n be the n-th triangle. How many dots are in the n-th triangle without counting? For

this example, suppose we want to count the number of dots in the 4th triangle.

If we add two 4th triangles, we get a rectangle that is 4× 5 with 20 dots. We know that the

4th triangle has 10 dots so we divide the number of dots in the rectangle by 2. In general,

to find the number of dots in the n-th triangle, we can use the following formula:

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Hockey Stick Pattern

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

If we look at a diagonal group of numbers starting at any of the bordering 1’s and ending

on any number inside the triangle, the sum of the numbers is equal to the number below

the end of the diagonal group which creates the image of a hockey stick on the triangle! As

shown above, we see that...

Prime Numbers

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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

Highlight each row where “term 1” is a prime number. Notice anything about the first term

and the numbers in its row? Each number within the triangle and row is a multiple of the

first term! Can’t see it yet? Let’s look more closely at row 7.

row 7 =⇒ 1 7 21 35 35 21 7 1

⇓ ⇓ ⇓ ⇓ ⇓ ⇓1× 7 3× 7 5× 7 5× 7 3× 7 1× 7

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Horizontal Sums

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 ⇒

The sum of the numbers in each row, the horizontal sum, is a power of 2! Also, the exponent

of each horizontal sum corresponds to its row number.

Row 4 =⇒ 1 + 4 + 6 + 4 + 1 =

Squares

For this pattern, we will looking more closely at the counting numbers in the second diagonal.

The square of a diagonal number is equal to the sum of the number beside it and the number

below. For example, as seen in the triangle below, we see that...

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

So we have that 52 = 25 = 10 + 15. Do you see any other squares?

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Powers of 11

This pattern is quite interesting. Let’s do a little experiment. Using a calculator (or use

mental math if you like), calculate 110, 111, 112, 113, and 114. What do you notice about

this numbers and the rows of Pascal’s Triangle?

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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

In this pattern, each row in Pascal’s Triangle is a power of 11! The row number is the

exponent. So how does this work? We read each term as a digit of a number as shown

above. But notice that this does not work for rows 5 onwards. We can still make the pattern

work! Here’s the trick (this applies to row 5 and beyond):

For each term with more than one digit, we keep the digit in the ones place value and carry

over the remaining digits to the next term.

1 5 10 10 5 1 =⇒ - - - - - 1

1 5 10 10 5 1 =⇒ - - - - 5 1

1 5 10 ← 10 5 1 =⇒ - - - 0 5 1

1 5 ← 11 0 5 1 =⇒ - - 1 0 5 1

1 6 1 0 5 1 =⇒ - 6 1 0 5 1

1 6 1 0 5 1 =⇒ 1 6 1 0 5 1

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In the example on the previous page, notice that we are working right to left.

- Start from the rightmost term. 1 is a single digit number so we keep it. Move on to

the next term.

- 5 is also a single digit number. Keep it and move on to the next term.

- 10 has two digits. Keep 0 in the ones place value and carry over 1 to the next term.

- 10 + 1 = 11. Now 11 has two digits. Keep 1 in the ones place value and carry over 1

to the next term.

- 5 + 1 = 6. 6 is a single digit number. Keep it and move on to the next term.

- 1 is a single digit number and our last term. Keep it and we are done.

- The resulting row is 161051 (which is equal to 115)!

Fibonacci Sequence

The Fibonacci Sequence is a well-known pattern of numbers introduced by Italian math-

ematician Leonardo of Pisa. His nickname is Fibonacci which means “son of Bonacci” in

Italian. The first two numbers are 1 and 1. To get the next number, we add the previous

two numbers together. Shown below is the beginning of the Fibonacci sequence.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, , , ...

We can also draw the Fibonacci sequence as a spiral of numbers. Mathematicians often refer

to this as the “golden spiral”.

11 2

35

8

13

21

34

55

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Now how is the Fibonacci Sequence found in Pascal’s Triangle? Start by circling the term

0 of any row. Next, circle the number that is one row above and one term over. Continue

circling numbers until we can no longer follow the pattern. If we add the circled numbers

together, the sum is a Fibonacci number! The first few Fibonacci numbers have been done

for you. Find the rest of the numbers yourself!

1 ⇒ 1

1 1 ⇒ 1

1 2 1 ⇒ 2 = 1 + 1

1 3 3 1 ⇒ 3 = 1 + 2

1 4 6 4 1 ⇒ 5 = 1 + 3 + 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Notice that the 8th term of the sequence begins at row 7, the 7th term begins at row 6, the

6th term begins at row 5, so on and so on. The n-th term begins at row (n - 1) of Pascal’s

Triangle.

Fun fact! Fibonacci was not the first person to know about the sequence! It was known in

India hundreds of years before.

Another fun fact! November 23 is celebrated as Fibonacci Day! Why? Written in the

mm/dd format, 11/23, the digits in the date form a Fibonacci sequence!

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Problem Set

1. Draw Pascal’s Triangle up until row 12.

2. Find the following numbers in Pascal’s Triangle:

a) row 8, term 8

b) row 10, term 7

c) row 12, term 6

d) row 13, term 9

3. Solve the following equations:

a) (row 8, term 8) + (row 10, term 7)

b) (row 12, term 7) - (row 10, term 7)

c) (row 5, term 2) + (row 11, term 1) - (row 6, term 3)

d) (row 10, term 3) - (row 8, term 4) + (row 7, term 5)

4. How many dots are in the following triangles of the Triangular Number Sequence?

a) 8th triangle

b) 10th triangle

c) 17th triangle

d) 20th triangle

5. What is the row number of 1287? Is the row number a prime?

6. What is the sum of the 11th row of Pascal’s Triangle?

7. Find the sum of the following and determine its row and term number in Pascal’s

Triangle:

a) 1 + 5 + 15 + 35 + 70 + 126

b) 45 + 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1

8. Using Pascal’s Triangle, find the squares of the following numbers:

a) 3

b) 6

c) 7

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9. The following is a portion of Pascal’s Triangle. Find the values of X, Y and Z:

1287 X

3003 3432 Y

6435

Z

10. What is the 15th term of Fibonacci’s sequence? .

*11. For this question, we are counting how many paths Bruce can take to get to each

destination if he can only travel down or to the right. We count the number of possible

paths he can take at each corner of the squares.

In the example below, there are 252 ways for Bruce to reach his destination.

(a) For the picture below, how many possible paths can Bruce take to reach the

forest?

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(b) How possible paths can Bruce take to reach the building? (Note that he can bike

along the black squares but he cannot bike through them.)

**12. Using Pascal’s Triangle, find the sum of

12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92

***13. Calculate 1113 using Pascal’s Triangle.

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