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## Pascal’s Triangle and Powers of 11

So, first, where can powers of 11 be found in Pascal’s triangle? If we look at the first row of Pascal’s triangle, it’s 1,1. We’re going to interpret it as 11. The second row is 1,2,1, which we call 121, 11×11, or 11 squares. Moving down to the third row, we get 1331, which is 11x11x11, or 11 cubed. 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 after line 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 format seems to have stopped working. However, we can actually apply this to row 5 and above, since we can interpret 1,5,10,10,5,1 as 161051.

First, we need to understand why the pattern seems to stop working – then we stand a chance of sorting things out. The reason is that in row 5, we suddenly have two digit numbers (10s). This is easier if we think of numbers that fit into spaces from Pascal’s triangle. In row 5, we are squishing two digits into the same space.

To understand how to interpret 1,5,10,10,5,1, we need to think about what we have been doing so far. When we see 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 to mean 1 in the units column. Now we can see. That when you get 10 in, for example the hundreds column, its actually 10x 100 = 1000. In other words, you treat ten as “0 carry 1” just like when you’re doing addition. This is shown below for 1,5,10,10,5,1:

1 5 0 0 5 1

+.1 1 These 1’s are taken from 10’s

= 1 6 1 0 5 1

Surprisingly, therefore, we can quickly calculate any power of 11 using Pascal’s triangle. This can sometimes help if you need to calculate powers of 11 quickly. However, the fun doesn’t stop here: by modifying Pascal’s triangle, we can quickly calculate any number multiplied by a power of 11. For example, we can calculate 241 x 11^2. All we do is start with 2,4,1 as our first row. When we try to multiply by 11^2, we have to calculate 2 more rows of Pascal’s triangle from this initial row. For this, we use the rules of adding the above two terms as in Pascal’s triangle. It 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

It’s a great way to quickly calculate sums involving multiplication 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 speed calculations!

To show why this works, let’s take the number abcd, (where a, b, c and d are each the digits 0 through 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 multiplying a number by 10, you add a 0 to the end of it, so abcd x 10 equals abcd0. Now, we can add this to abcd x 1:

abcd 0

+. A B C D

This gives the answer a(+0) b+a c+b d+c 0+d. It may seem unnecessary, but wait a minute! This is exactly like the sum of Pascal’s triangle! You can check this using another diagram.

… A B C D

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

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

The same process can be applied for any numbers. So, we can see why this clever little trick works, although that doesn’t make it any less spectacular and still definitely worth trying out on your friends!

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