Pascal’s Triangle

This pattern is named after the French mathematician Blaise Pascal (1623-62) who brought the triangle to the attention of Western mathematicians (it was known as early as 1300 in China, where it was known as the "Chinese Triangle").The triangle itself is made by arranging numbers.

This picture shows the first five lines of Pascal’s triangle. Can you work out how it is made?

Each number is the total of the two numbers above it. Work out the next five lines of Pascal’s triangle and write them below.

- Look at your diagram. What patterns can you see?

- Look at the odd numbers. How are they arranged in the triangle?

- Now, look at the even numbers. How are they arranged in the triangle?

- Look at the diagonal lines on the triangle. What patterns do you notice?

- Is there a way of predicting the next line of the triangle, without having to work out each number by adding the two numbers above it?

- Investigate the totals of the numbers in each horizontal row. Is there a pattern? Can you predict the next total?

Patterns Within the Triangle

/ Diagonals
The first diagonal is, of course, just "1"s, and the next diagonal has the Counting Numbers (1,2,3, etc).
The third diagonal has the triangular numbers
(The fourth diagonal, not highlighted, has the tetrahedral numbers.)

Triangular Number Sequence

This is the Triangular Number Sequence:

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

This sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

Tetrahedral Number Sequence

The Tetrahedral Number Sequence can be easily understood if you think of a stack of marbles in the shape of a Tetrahedron.
Just count how many marbles would be needed for a stack of a certain height.
·  For height=1 you only need one marble
·  For height=2, you would need 4 marbles (1 at the top and 3 below)
·  For height=3 you would need 10 marbles.
·  For height=4 you would need 20 marbles.
·  How many for height=5 (like the illustration) ... ? /

Triangular and Tetrahedral Numbers

Each layer in the tetrahedron of marbles is actually part of the Triangular Number Sequence (1, 3, 6, etc). And both the triangular numbers and the tetrahedral numbers are on Pascal's Triangle.

This table shows the values for the first few layers:

n / Triangular Number / Tetrahedral Number
(Height) / (Marbles in Layer) / (Total Marbles)
1 / 1 / 1
2 / 3 / 4
3 / 6 / 10
4 / 10 / 20
5 / 15 / 35
6 / 21 / 56

If you look at the numbers you can see something interesting: if you take any number and add the number below and to the left, you get the next number in the sequence. (For example 6+4=10).

Odds and Evens
If you color the Odd and Even numbers, you end up with a pattern the same as the Sierpinski Triangle /
/ Horizontal Sums
What do you notice about the horizontal sums?
Is there a pattern? Isn't it amazing! It doubles each time (powers of 2).
Exponents of 11
Each line is also the powers (exponents) of 11:
·  110=1 (the first line is just a "1")
·  111=11 (the second line is "1" and "1")
·  112=121 (the third line is "1", "2", "1")
·  etc!
But what happens with 115 ? Simple! The digits just overlap, like this:

The same thing happens with 116 etc. /
Fibonacci Sequence
Try this: make a pattern by going up and then along, then add up the values (as illustrated) ... you will get the Fibonacci Sequence.
(The Fibonacci Sequence starts "1, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc) /
/ Symmetrical
And the triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.