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 just the two numbers above it added together (except for the edges, which are all "1").
(Here I have highlighted that 1+3 = 4) 



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.) 
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). 
Fibonacci Sequence
Try this: make a pattern by going up and then along, then add up the squares (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. 

Using Pascal's Triangle
Heads and Tails
Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you "the odds" (or probability) of any combination.
For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three that will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern "1,3,3,1" in Pascal's Triangle.
Tosses 
Possible Results (Grouped) 
Pascal's Triangle 
1 
H
T 
1, 1 
2 
HH
HT TH
TT 
1, 2, 1 
3 
HHH
HHT, HTH, THH
HTT, THT, TTH
TTT 
1, 3, 3, 1 
4 
HHHH
HHHT, HHTH, HTHH, THHH
HHTT, HTHT, HTTH, THHT, THTH, TTHH
HTTT, THTT, TTHT, TTTH
TTTT 
1, 4, 6, 4, 1 

... etc ... 


What is the probability of getting exactly two heads with 4 coin tosses?
There are 1+4+6+4+1 = 16 (or 2^{4}=16) possible results, and 6 of them give exactly two heads. So the probability is 6/16, or 37.5% 
Combinations
The triangle also shows you how many combinations of objects are possible.
For example, if you have 16 pool balls, how many different ways could you choose just 3 of them (ignoring the order that you select them)?
Answer: go down to row 16 (the top row is 0), and then along 3 places and the value there is your answer, 560. Here is an extract at row 16:
1 14 91 364 ...
1 15 105 455 1365 ...
1 16 120 560 1820 4368 ...
Polynomials
Pascal's Triangle can also show you the coefficients in binomial expansion:
Power 
Polynomial Expansion 
Pascal's Triangle 
2 
(x + 1)^{2} = 1x^{2} + 2x + 1 
1, 2, 1 
3 
(x + 1)^{3} = 1x^{3} + 3x^{2} + 3x + 1 
1, 3, 3, 1 
4 
(x + 1)^{4} = 1x^{4} + 4x^{3} + 6x^{2} + 4x + 1 
1, 4, 6, 4, 1 

... etc ... 

The First 15 Lines
For reference, I have included row 0 to 14 of Pascal's Triangle
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
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

The Chinese Knew About It
This drawing is entitled "The Old Method Chart of the Seven Multiplying Squares". View Full Image
It is from the front of Chu ShiChieh's book "Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements), written in AD 1303 (over 700 years ago!), and in the book it says the triangle was known about more than two centuries before that. 
The Quincunx

An amazing little machine created by Sir Francis Galton is a Pascal's Triangle made out of pegs. It is called The Quincunx.
Balls are dropped onto the first peg and then bounce down to the bottom of the triangle where they collect in little bins.
At first it looks completely random (and it is), but then you find the balls pile up in a nice pattern: the Normal Distribution. 


