PASCAL'S TRIANGLE

Pascal's Triangle

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern.

Each number is the numbers directly above it added together.


(Here I have highlighted that 1+3 = 4)

Patterns Within the Triangle

Diagonals

The first diagonal is, of course, just "1"s

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.)

Symmetrical

The triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.

Horizontal Sums

What do you notice about the horizontal sums?

Is there a pattern?

They double 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!

Squares

For the second diagonal, the square of a number is equal to the sum of the numbers next to it and below both of those.

Examples:

  • 32 = 3 + 6 = 9,
  • 42 = 6 + 10 = 16,
  • 52 = 10 + 15 = 25,
  • ...

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 "0, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

Odds and Evens

If we colour the Odd and Even numbers, we end up with a pattern the same as the Sierpinski Triangle

Paths

Each entry is also the number of different paths from the top down.

Example: there is only one path from the top down to any "1"

pascals triangle path to 1

And we can see there are 2 different paths to the "2"

pascals triangle paths to 2

It is the same going upwards, there are 3 different paths from 3:

pascals triangle paths to 3

Your turn, see if you can find all the paths down to the "6":

pascals triangle paths

YouTube:Link:https://youtu.be/XMriWTvPXHI