Pascal’s Triangle

This is Pascal’s Triangle. Do you see the pattern determining the next row? Basically, you add the two numbers above to get the next entry. So the next line would read: 1, 7, 21, 35, 35, 21, 7, 1.

For those who like math notation:

The formula for nCr is:
n!
——–
r!(n-r)!

(The ! means factorial, aka: 5! = 1 x 2 x 3 x 4 x 5 = 120. It’s super useful in basic statistics, but that’s another entry.)

The weird thing about this is not how you make the pattern, but that the pattern is actually useful!

The sum of the rows gives the powers of 2.
Row 0: = 1 = 20
Row 1: = 1 + 1 = 2 = 21
Row 2: =  1 + 2 + 1 = 4 = 22
Row 3: = 1 + 3 + 3 + 1 = 8 = 23 …

If the second number in the row is prime, (because the first number in all the rows is 1, so skip it) the all of the numbers in that row are divisible by it. Check the 8th row that I wrote out up top as a good example:

1, 7, 21, 35, 35, 21, 7, 1

Something called the Magic 11’s:
If each row is a single number, it is a power of 11:
1 = 110
11 = 111
121 = 112
1331 = 113
14641 = 114  …

Here’s my favorite one. So let’s say you are trying to solve (x + y)n for some variables x and y, raised to the power n.
(x + y)0 = 1
(x + y)1 = x + y       aka 1x + 1y
(x + y)2 = x+ 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy+ y3
(x + y)= x4 + 4x3y + 6x2y+ 4xy+ y4

The coefficients are the entries in the triangle! Crazy.

There’s a whole bunch more that’s probably only interesting to people who do math alot, but it’s pretty damn nifty. Check out http://ptri1.tripod.com/ for more.

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