More Patterns at CTY: All in one triangle!!

This week is my second week at Johns Hopkins CTY program. We did a really interesting class on Pascal's Triangle and it's beautiful properties. I loved them and would like to share some with you.

First off: 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

If you look, each number is generated by adding the number above it to the number to the left of the one above it. For instace, the 20 in the last row is generated by adding the 10 above it and the 10 to the left of the number above it.

One pattern is found by adding up the numbers in each row. Check it out:

1 = 1
1 + 1 = 2
1 + 2 + 1 = 4
1 + 3 +  3 + 1 = 8
1 + 4 +  6 + 4 + 1 = 16

It's pretty obvious! It's the powers of two! 2^0, 2^1, 2^2, and so on. I think that is pretty cool, right?

Also, I had noticed that each row written as a number has a pattern not as obvious, but still cool. Check it out:

1 = 11^0
11 = 11^1
121 = 11^2
1331 = 11^3
14641 = 11^4

This also continues, and you will understand why in a future post.

Bonus: Pascal Magic

Though I wrote it so I could type it easily, the triangle is generally written differently. If you google it, you'll see what I mean. Anyways, have someone draw a rectangle around the top one and make it as small or big as they want. Then, tell them to add up all the numbers in the rectangle and you can tell them the answer immediately.

All you need to do is subtract one from the number directly under the bottom of the rectangle.

Challenge: If you were to begin the triangle with say two instead of one and did the trick, how do you find the sum? What about three? Or 100?