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## Pascal’s Triangle and Powers of 11

So, first of all, where do you find the powers of 11 in Pascal’s triangle? If we look at the first row of Pascal’s triangle, it is 1.1. We’ll interpret that as 11. The second row 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 cubes. And from the fourth row, 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 row 5? Row 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 have stopped working. However, we can apply it to rows 5 and beyond, since we can interpret 1,5,10,10,5,1 as 161051.

First, we need to understand why the pattern seems to have stopped working; then we have a chance to work things out. The reason is that in row 5, we suddenly got two-digit numbers (the 10s). It’s easier if we think of the numbers in Pascal’s triangle that fit into the spaces. In row 5, we are popping two digits into the same space.

to understand how to interpret 1,5,10,10,5,1, we have to think about exactly what we’ve been doing 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 10 in, say, the hundreds column, that really means 10x 100 = 1000. In other words, you just treat the ten as “0 leads to 1” like when you do addition. This is shown for 1,5,10,10,5,1 below:

1 5 0 0 5 1

+.1 1 these 1’s are taken from 10’s

= 1 6 1 0 5 1

Amazingly, therefore, we can quickly calculate any power of 11 using Pascal’s triangle. This can occasionally help 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 row. Since 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 adding the two previous terms just like in Pascal’s triangle itself. This is shown below:

2,4,1

2,6,5,1

2,8,11,6,1

2 8 1 6 1

… 1

2 9 1 6 1

This is a great way to calculate sums that involve multiplying by 11 quickly, so even if you’ve never been good at arithmetic, try it on your friends or family and impress them with your lightning speed calculations!

To show why this works, let’s take the number abcd, (where a, b, cid are each a digit from 0 to 9) and multiply it by 11. We can split 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 just add a 0 to the end, so abcd x 10 is the same as abcd0. Now, we can add this to abcd x 1:

abcd 0

+. a B C D

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

… a B C D

(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. So we can see why this clever little trick works, although that doesn’t make it any less awesome and worth trying out with your friends!

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