Pascal's+Triangle


 * __The Pascal's Triangle__**



In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy.

__**Finding the numbers**__

We re able to use a formula known as the recursive formula to find the positions of numbers in a Pascal's Triangle. Although it is slightly redundant, it is a way to show how you derive the two numbers that add up to form the number below. Note that the [brackets] represent subscripts.
 * t[n,r] = t[n-1,r-1] + t[n-1,r]**

Where //n// is the row number and //r// is the position on the row. Also note that the __first position and row is considered position 0__.

For example: How do we find which two numbers make t[9,1]? t[9,1] is basically the real row 10 and real position 2. Sub t[9,1] into the equation. t[9,1] = t[9-1,1-1] + t[9-1,1] = t[8,0] + t[8,1]

Also, it is a given rule that for every row //n//, 2^n is the number you get when you add all the numbers in the row together.

Row 0 = 2^0 = 1 Row 1 = 2^1 = 2 Row 2 = 2^2 = 4 Row 3 = 2^3 = 8

And so on and so forth......

For example, find which row in the Pascal's Triangle has integers with a total sum of 1024. 2^n = 1024 n log 2 = log 1024 n = log 1024 / log 2 n=10

Thus row 10 is the answer. (Real row 11)

Pascal's version of his triangle. Illustration from book "Precious Mirror" from 1303 AD by Chinese mathematician Zhu Shijie.
 * __Pictures (From left to right):__**



The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called Sierpinski triangle, and this resemblance becomes more and more accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern //is// the Sierpinski triangle, assuming a fixed perimeter. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other patterns and combinations.
 * __Patterns in the Pascal's Triangle__**

__Picture: A Sierpinski Triangle (Fractal)__



__Picture: Four Different Cases of Isolating Certain Integers in the Pascal's Triangle__


 * Note:** A fractal is a shape that is "infinite", meaning you can zoom in forever but the shape will never end.

__This is a very powerful and interesting site to explore fractals.__ []

Pascal's Triangle can be used to find how many ways can you reach a particular place. It can calculate possibilities very efficiently.
 * __Applications of the Pascal's Triangle__**

For example: How many ways can you spell pascal with the diagram below? From P, you can split into the two As. From the left A, you can split to the left S, from the right A, you can split to the right S, and from both A's you can split ro the middle S. Thus, there are (1+2+1)=4 possible ways up till this moment.

With the below Pascal's "shape", you will eventually notice that the answer for this example is 10+10=20, the sum of the two integers at the bottom of the diagram.

Another example: If a checkerpiece can move diagonally upwards on the board, to the left or right, but cannot move onto the square with the "X", how many possible solutions are there?

Again, the best way to solve this is to draw a Pascal's Shape, and then find the sum of the last row of integers (In this case, the top row).