Pascal's triangle II

Last time, we looked at Pascal's Triangle, which is what you get if you start with a 1, and then make new rows by adding the numbers above, like this:

We saw that Pascal's Triangle could give us the answers to questions like how many ways we could choose ice cream flavours or an opening batting pair. This time, we're going to look at how Pascal's Triangle works and also at some hidden patterns in it.

First of all, let's see why this trick of adding the numbers above a certain space gives us the answer, so we don't have to do those boring lists we had to do last time. Let's check out an example. Suppose we have 4 flavours of ice cream to choose from, and we want to choose 2 of them. According to the triangle, this gives us 6 possibilities (remember, the triangle has a 0th row and a 0th box on each row). The 6 came from the two 3's above, which were the ways to choose 1 flavour from 3, and 2 flavours from 3. Why would adding those together give us how many ways we could choose 2 from 4? Well, we can break the number of ways we can choose 2 from 4 into two parts. Let's say the flavours are chocolate, vanilla, strawberry, and mango. Either chocolate is one of the flavours we choose, or it isn't. Those are our two parts. If chocolate is one of our flavours, then for the other flavour, we have 3 choices left. Thus, we have to choose 1 flavour from 3. That's one of the boxes in Pascal's Triangle. What about the other box? Well, if chocolate isn't one of the flavours we choose, then we need to choose 2 flavours from the 3 remaining ones. That's the other box. This covers all the possibilities, since any possibility either has chocolate or it doesn't (just like it's either raining or not raining).

We can do this for any box in the Triangle. Any box is talking about the number of ways to choose some number of flavours from some total number. To figure out what goes in that box, we can say that either some flavour Q is going to be in our choice or it isn't. If it is, then look at the box above to the left. If it isn't, then look at the box above to the right. Together, those boxes give all the possibilities. So that's why the Triangle works.

There's one thing I've left out - the 1's along the sides. They don't seem to make any sense. Now, remember, since the numbering starts at 0, the 1 in the 2nd row on the left is saying that if you choose 0 out of 2 flavours, there is one way to do that. This may seem silly, but it makes sense. Imagine an ice cream cone. On it, you can put scoops of ice cream. If you have money for only 1 scoop, then if there are 2 flavours, you have 2 choices. If you have no money, then no matter what, all you'll have is an empty cone. That's your only possibility. On the other hand, if you have the money and intend to get 2 scoops, then since there are only 2 flavours, you have only one possibility there too. That explains the second row. The same reasoning works for all the other rows, except the 0th row. Here, the ice cream man tells you that he's out of ice cream. Then you have only one choice - not to have any ice cream.

That finishes the explanation of why the triangle works. Now, let's take a closer look at one of its cool properties - the pattern of odd and even numbers. Often in maths, when we want to find patterns we have to get rid of some useless information which is just getting in the way. Looking at Pascal's Triangle, it's hard to see a pattern of odds and
evens, but that's because the numbers are getting in the way. We can get around this problem by getting rid of the numbers. Colour a box white if the number in it is odd, and black if it's even. Now, since two even numbers added together make another even number, even + odd = odd, and odd + odd = even, we don't even have to write down the numbers - just start with a white box at the top, and colour away. If you have squared paper, you should be able to make a 20-row Triangle without too much trouble. See a pattern? Here's a very big Triangle (255 rows) coloured in this way:

Now the pattern is very clear, right? At this scale, it looks like we have a big white triangle. In the middle of that triangle, we colored an upside-down triangle black. That left us with 3 smaller white triangles. Repeat for those triangles, and repeat and repeat . . . An object that looks like this, with smaller and smaller copies inside it is called a fractal. This particular fractal is called a Sierpinski Gasket. Of course, we know that Pascal's Triangle is made of boxes, so the copies don't keep on getting smaller and smaller, but we can think of it as going the other way - as Pascal's Triangle gets bigger and bigger, it looks the same, since it's just made out of copies of itself. (see the article
On fractals for how to make it yourself)

This gives a very nice way to make the Sierpinski Gasket, but why does it work? Why does Pascal's Triangle do this? Let's take a small piece of the Triangle to start with - just the top 3 rows (4 with the 0th row). This triangle's bottom row is all odds. This means that the next row is going to be all evens, except at the ends, where the odds are always there. But remember, odd + even = odd. This means that those even squares, the black squares, might as well not be there. For those two odds at the ends, it's exactly like starting their own new Pascal's Triangle; the black squares are just like the blank squares around the top white square - they don't matter. Except that as those two triangles expand, they eat up the black in between them, until they meet. There were 3 black squares to start with. They're all gone after 3 more rows. But in those 3 rows, the 2 end odds have created 2 copies of that original small triangle. Since its bottom squares were all white/odd, the bottom row now is also all white. But now we have a bigger triangle with the same property - that its bottom row is all white. We can reason exactly the same way: the next row will be all black except for the ends, which will each generate their own copy of this bigger triangle, until they meet, making an even bigger triangle. This keeps on going. Thus, to make Pascal's Triangle, we don't even have to remember the adding rules for black and white. Once we've made that first small triangle, we can just stick three copies of it around a black triangle, then take this new bigger triangle and stick three copies of it around a black triangle, etc.

Finally, there's no reason we need to restrict ourselves to even and odd numbers. We could color squares based on whether they were multiples of 3, 1 more than a multiple of 3, or 2 more than a multiple of 3. You get very cool designs for higher numbers - try it!

-Janak Ramakrishnan