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Pascal’s triangle

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Where’s the Maths?

1

3

6

10

15

21

square numbers

1

4

9

16

25

36

pentagonal numbers

1

5

12

22

35

51

hexagonal numbers

1

6

15

28

45

66

Eugène Charles Catalan defined the Catalan Numbers as the number of ways to disect a polygon with non intersecting diagonals.

Catalan numbers

Catalan numbers number of ways to disect a polygon with non intersecting diagonals.

Catalan numbers number of ways to disect a polygon with non intersecting diagonals.

1 number of ways to disect a polygon with non intersecting diagonals.

1

2

3

5

8

13

Fibonacci numbers

prime numbers number of ways to disect a polygon with non intersecting diagonals.

odd numbers number of ways to disect a polygon with non intersecting diagonals.

even numbers number of ways to disect a polygon with non intersecting diagonals.

Multiples of 3 number of ways to disect a polygon with non intersecting diagonals.

Multiples of 4 number of ways to disect a polygon with non intersecting diagonals.

Multiples of 5 number of ways to disect a polygon with non intersecting diagonals.

Multiples of 6 number of ways to disect a polygon with non intersecting diagonals.

Multiples of 7

Is it a cube? number of ways to disect a polygon with non intersecting diagonals.

Combining triangles

What happens if I…. number of ways to disect a polygon with non intersecting diagonals.

Use a different number at the top of the triangle?

Use different numbers down the sides of the triangle?

Subtract the numbers instead of adding them?

Multiply the numbers instead of adding them?

Use fractions?

Use decimals?

What happens if I…. number of ways to disect a polygon with non intersecting diagonals.

Look at the triangle from a different angle?

Harmonic triangle number of ways to disect a polygon with non intersecting diagonals.

Yang number of ways to disect a polygon with non intersecting diagonals.hui’s triangle

Pascal’s petals number of ways to disect a polygon with non intersecting diagonals.

Pascal petals number of ways to disect a polygon with non intersecting diagonals.

Antonio’s Pizza Palace number of ways to disect a polygon with non intersecting diagonals.

It's Friday night and the Pizza Palace is more crowded than usual. At the counter the Pascalini's are trying to order a large pizza, but can't agree on what topping(s) to select.

Antonio, behind the counter, says, "I only have 8 different toppings. It can't be that hard to make up your mind. How many different pizzas could that be?"

"Well, we could get a plain pizza with no toppings," says Mr. Pascalini.

"Or we could get a pizza with all 8 toppings," says Mrs. Pascalini.

"What about a pizza with extra cheese and green peppers?" asks Pepe.

"You're not helping!" Antonio yells at Pepe. "Get back to work."

As Pepe starts to clear off the nearest table, he mumbles to himself, "or a pizza with anchovies, extra cheese, mushrooms, and olives."

Antonio hands an order pad to Mr. Pascalini and says, "When you decide, write it down and I'll make it."

Then he helps the next people in line, who know what they want: a large pizza with mushrooms, green peppers and tomatoes.

How many different pizzas can be ordered at the Pizza Palace if a pizza can be selected with any combination of the following toppings:

anchovies, extra cheese, green peppers, mushrooms, olives, pepperoni, sausage, and tomatoes?

How many different pizzas can you order with only one topping?

How many different pizzas can you order each with seven toppings?

Are the number of one-topping pizzas and the number of seven-topping pizzas related? (Why or why not?)

How many different pizzas can you order with two toppings?

How many different pizzas can you order with six toppings?

Are the number of two-topping pizzas and the number of six-topping pizzas related? (Why or why not?)

Can you find these numbers in Pascal's triangle?

Can you use Pascal's triangle to help you find the number of pizzas that can be ordered with three, four, or five toppings?

In all, how many different pizzas can be ordered?

Points on a circle topping?

sierspinski topping?

Is there a way to predict the number of downward triangles you will have after n iterations?

Twelve days of topping?christmas

On the twelfth day of Christmas my true love sent to me

Twelve drummers drumming, Eleven pipers piping,

Ten lords a-leaping, Nine ladies dancing,

Eight maids a-milking, Seven swans a-swimming,

Six geese a-laying, Five gold rings, Four calling birds,

Three French hens, Two turtle doves

and a partridge in a pear tree.

City grid topping?

Starting in the top left corner of a 2 x 2 grid, there are 6 routes

(without backtracking) to the bottom right hand corner.

How many routes are there through a 20 x 20 grid?

Heads and tails topping?

How many different combinations are there when you flip a coin once?

How many different combinations are there when you flip a coin twice?

How many different combinations are there when you flip a coin ten times?

Heads and tails topping?

Blaise topping?pascal

Blaise Pascal was a French mathematician who lived from 1623-1662.

A topping?pascaline