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Pascal's Triangle on Scorpion night before last: Math Tower Activity

Pascal's Triangle was on Scorpion the night before last. After a jail break Walter finds himself tracking down, or at least trying to connect with the jail breakers. His team is with him.

Walter is played by Elyes Gabel. But, most importantly is what happens next. In an effort to meet the runaway jail breakers, Walter works with his side kick, the human computer, to create an engaging task that will lure the smart one of the jail breakers into connecting over the internet.

The task is to decipher some puzzle related to the Pascal's Triangle. "Too easy and we are flooded by math geeks, too hard and we don't get him", Happy basically says.

This is Pascal's Triangle:

1
121
1331
14641

and so on. You may or may not know that these numbers as above are the coefficients of (x+1)n, which is a polynomial of degree n. If I were to rewrite the Pascals Triangle to display this relationship I would get:

1
1x+1
1x2+2x+1
1x3+3x2+3x+1
1x4+4x3+6x2+4x+1

Ignore the exponents and just focus on the coefficients. You see the connection!

Its just like the human computer says, each item in the row, except the sides which are all one, is the sum of the two that are above it. This can be easier to see on a well centered chart of the numbers.

If you like Pacal's Triangle, you'll like this activity. We are going to build towers and draw paths that cover the conceptual territory of the Triangle in question. To investigate:

Start by drawing a grid of 2x2 = 4 squares! on your page. Then draw a similiar 3x3. Draw a pathway through the squares connecting adjacent squares. My 3x3 looks like this:

OXX
XXO
XOO

I have created a path of Xs that go through the Os. How many paths are possible on a 2x2 grid? There are only two, we start at the bottom left and end at the top right. The game is to count pathways:

1.
OX
XX

2.
XX
XO

In case you are having a hard time visualizing these examples, think of a frog that hops from stepping stone to stepping stone. The Xs mark its path along each stone. The stones are the grid squares. To continue, I leave it up to you the reader to make another 3x3. How many paths are there? It turns out this is a combinatorial function. Its represented by "c" on your calculator. C is for choose.

We come back to the calculator at the end of this entry. First, lets make a tower. For each and every one of those paths, we can make a tower which encapsulates, encodes, embodies, emphasizes, captures, whatever you want to call it, the path, and the info it contains.

By cutting apart some paper we can see the intricacies of Pascal's triangle.

1. Cut an 8.5x11 into 9 equal pieces. Each piece should be the size of a playing card.

2. Out of the 9 make 3 to be medium size, 3 to be large size, and make 3 to be small size. They are the levels of the tower. The small should be quite small and the medium size is in the middle. You may wish to square off the 9 pieces.

4. In each group of three label a 1, a 2, and a 3.

You should have 9 pieces of paper:
(L,3)(M,3)(S,3)
(L,2)(M,2)(S,2)
(L,1)(M,1)(S,1)

5. Assemble towers with 5 out of the 9 tiles so that every consecutive level either repeats in size or in number and the numbers and sizes get larger, like in a real tower. Think of these repeats as the glue that bonds the levels. The papers must connect for a valid tower.

Here is my tower:

(L,1),(L,2),(L,3),(M,3),(S,3)

This is not a valid part of a tower:

(M,2)(S,3)

because both the sizes of the paper and the number increased at the same time.

If you have ever played a game called Towers of Hanoi, these towers should have a similiar look to them.

Take a look at our pascals triangle again, only this time in the form of a grid:

136
123
111

Because of the correspondence between paths and towers captured, my tower is actually a path leading towards 6.

XXX
X23
X11

There is a huge thing happening here. The labeling exactly captures the number of paths up to that point, and the number of towers that can be made.
 
Finally we conclude with by going back to the calculator. The total number of towers is equal to "4 choose 2". 4 is the diagonal distance. 2 is the distance from the side of pascal's triangle counting up from zero. This way works on the calculator to count all the sets of much larger towers.
 

$20p/h

Elias H.

Patient and Knowledgeable Liberal Arts Math Tutor

100+ hours
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