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## Pascal’s Triangle and Powers of 11

So, first, where to find the powers of 11 in Pascal’s triangle? If we look at the first row of Pascal’s triangle, it’s 1.1. We’ll interpret that as 11. The second line is 1,2,1, which we’ll call 121, which is 11×11, or 11 squared. Going down to the third row, we get 1331, which is 11x11x11, or 11 cubed. And from the fourth line we get 14641, which is 11x11x11x11 or 11^4. This information is summarized in the diagram below:

1 1

1 2 1

1 3 3 1

1 4 6 4 1

11 = 11^1

121 = 11^2

1331 = 11^3

14641 = 11^4

But what do we do from rank 5? Line 5 is 1,5,10,10,5,1, but if you have a calculator you can check that 11^5 is 161051, not 15101051. The pattern seems to stop working. However, we can actually apply it to lines 5 onwards, because we can interpret 1,5,10,10,5,1 as 161051.

First, we need to figure out why the model seems to have stopped working – then we have a chance to fix things. The reason is that in line 5 we suddenly have two digit numbers (the 10s). It’s easier if we think that the numbers in Pascal’s triangle fit into spaces. In row 5, we overwrite two digits in the same space.

to understand how to interpret 1,5,10,10,5,1, we need to think about exactly what we have done so far. When we saw 1,2,1, for example, we put the first 1 in the hundreds column to mean 100, the two in the tens column to mean 20, and the last 1 in the units column to mean 1 Now we can see that when you get a 10 in, say, the hundreds column, it actually means 10x 100 = 1000. In other words, you just treat the ten as “0 carries 1” like when you make an addition. This is shown for 1,5,10,10,5,1 below:

1 5 0 0 5 1

+.1 1 these 1 were raised from 10

= 1 6 1 0 5 1

Amazingly, then, we can quickly calculate any power of 11 using Pascal’s triangle. This can help occasionally if you ever need to calculate a power of 11 quickly. However, the fun doesn’t stop there: by modifying Pascal’s triangle, we can quickly calculate any number multiplied by a power of 11. For example, we could calculate 241 x 11^2. All we do is start with 2,4,1 as the first line. As we are trying to multiply by 11^2, we need to calculate 2 more rows of Pascal’s triangle from this initial row. For this, we use the rules of addition of the two terms above, just like in Pascal’s triangle itself. This is illustrated below:

2,4,1

2,6,5,1

2,8,11,6,1

2 8 1 6 1

… 1

2 9 1 6 1

It’s a great way to quickly calculate sums by multiplying by 11, so even if you’ve never been good at arithmetic, try it out on your friends or family and impress them with your lightning-fast calculations. ‘flash !

To show why this works, take the number abcd, (where a, b, c, and d are each a digit from 0 to 9), and multiply it by 11. We can divide this multiplication into two bits, as in the diagram below :

abcd x 11=abcd x 10 + abcd x 1

When you multiply a number by 10, you simply add a 0 to the end of it, so abcd x 10 is the same as abcd0. Now we can add this to abcd x 1:

ABC 0

+. a B cd

This gives an answer of a(+0) b+a c+b d+c 0+d. This may seem unwieldy, but wait a minute! This is exactly the same as Pascal’s triangle sums! You can verify this using the following diagram.

… a B cd

(0+)a a+b b+c c+dd(+0)

=a(+0) b+a c+b d+c 0+d

A similar process can be applied for any number of digits. Therefore, we can see why this clever little trick works, even if it doesn’t make it any less spectacular and still worth trying on your friends!

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