## Saturday, October 13, 2012

### 100th Post: The First Secret of Mental Mathematics

I am thrilled to announce that today marks the 100th post on Cool Math Stuff. It looks like this will also be the longest post, since I have so much I want to say. Anyways, it is really great to be able to talk about what I think makes math cool. But I don't want it to just be what I think, I want you guys to give your input as well. So, please feel free to comment, or send me anything that you find cool about math. I will be more than happy to post about it (it saves me from having to think of something cool to write about).

Before I go into my post, I wanted to start finding out what you think is interesting in math. So, I want you to comment with the most interesting post I have written (the title or link will be fine), and I will take the one with the most comments and either repost it or add some input to it. Here are my top ten favorites, which you can feel free to choose from, or find one of your own.

#1. April 14, 2012; Probability, the Number e, and Magic all in one

#2. August 27, 2011; Do the Primes go to Infinity and Beyond?!!

#3. July 9, 2011; Divide Almost Any Odd Number into a Number Consisting of all Nines

#4. March 24, 2012; Pi, Lie, Same Thing... (Part Two)

#6. October 15, 2011; Why Does 64 = 65? Or does it...

#7. June 25, 2011; Greatest Common Factor Made Easy: It's Euclid to the Rescue

#8. September 17, 2011; What does .99999... Really Mean?

#9. July 23, 2011; Patterns and Puzzles at CTY

#10. January 14, 2012; Can you correctly add six numbers?

Since this is the 100th post, I wanted to make it something that is really special. And in fact, it is what pulled me into the fascinating aspects of mathematics.

In February of 2010, I was shown a video on TED of math professor and mathemagician Dr. Arthur Benjamin, who was doing incredible feats of mental math in his head. As a curious ten-year-old, I of course wanted to be able to do it too. So, my dad bought me his book, Secrets of Mental Math (there is now a fantastic video course on it, which you can click here to see), and I was slowly pulled into the whole mathematics, magic, and science community.

But those first few pages were what really drew me in. So, I wanted my hundredth post to be about what those first few pages were on: how to multiply by eleven in your head.

First of all, lots of you probably remember from third grade when you learned the following pattern:

1 x 11 = 11
2 x 11 = 22
3 x 11 = 33
4 x 11 = 44
5 x 11 = 55
6 x 11 = 66
7 x 11 = 77
8 x 11 = 88
9 x 11 = 99

That's always a fun one. But there still is a pattern that continues. Let's move it up to two-digit numbers.

10 x 11 = 110
11 x 11 = 121
12 x 11 = 132
13 x 11 = 143
14 x 11 = 154
15 x 11 = 165
16 x 11 = 176
17 x 11 = 187
18 x 11 = 198

This one is a little harder to see. Let me make it a little easier to see.

10 x 11 = 110
11 x 11 = 121
12 x 11 = 132
13 x 11 = 143
14 x 11 = 154
15 x 11 = 165
16 x 11 = 176
17 x 11 = 187
18 x 11 = 198

You might notice that going down the list without the bold gives you 123456789. But the important part is the bold. What numbers do you see?

The numbers we were multiplying by! So, the first digit is the first number and the second digit is the second number. That takes care of a lot of it. But what about the middle digit? Do you see any pattern between the outside numbers and the inside number?

1 + 0 = 1
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
1 + 7 = 8
1 + 8 = 9

The two outside digits actually sum to the middle one. Let's try a couple. 26 x 11.

2 + 6 = 8
286

6 + 1 = 7
671

Try 72 x 11.

7 + 2 = 9
792

3 + 9 = 12
3129

Wait, how can that be? 39 x 11 can't be bigger than 72 x 11. We must have messed up.

I neglected to mention before that there is only room for one digit in the middle spot. Since the full 12 doesn't fit there, the two drops in and the one carries over to the 3, making the answer 429.

1
329
429

What is 67 x 11?

1
637
737

Try 96 x 11. This might be confusing, but stick to our rules.

1
956
1056

Let's look at three digit numbers. Do you see any patterns with these ones:

132 x 11 = 1452
254 x 11 = 2794
816 x 11 = 8976
427 x 11 = 4697
354 x 11 = 3894

For the first one, you see the one and two on the outsides, just like the two-digit numbers. But where is the three? What are the four and the five for?

Well, 1 + 3 = 4, and 3 + 2 = 5. This goes for all of the other ones.

How about we try a couple:

1  5 3
\ / \ /
16  83

153 x 11 = 1683

7  2  4
\ / \ /
79 64

724 x 11 = 7964

8  4  2
\ /  \ /
8      2
12  6_
92  62

9    8     7
\   /  \   /
9           7
1(7+1)5
10857

That last one was a little confusing, but you will get the hang of it with some practice. And you can even take this up to four-digits, fives, and more. Let me quickly go over why this works. Let's just do 15 x 11 with classic multiplication from grade school.

15
x  11

So, 1 x 5 is five.

15
x  11
5

1 x 1 = 1. Remember to put the zero down as a place holder.

15
x  11
15
0

Now, 1 x 5 is 5. 1 x 1 is 1.

15
x  11
15
150

15
x  11
15
+150
165

That is great. But, what happened in the process? On the left end, you have just the one (from the one in the fifteen). On the right, you have just a five. In the middle, you have the one and the five, which is the same as adding the two digits together and putting them in the middle.

I find the proof of this pretty cool, but I especially find it cool that you can multiply a number by eleven in seconds with a day or two of practice.

Bonus: I don't normally do a bonus on a long post, but this is something I would really like to share. Recently, I was thrilled to receive an email from a reader of the blog, who wanted to know a formula for figuring out a problem like so (this is his exact words in the email):

"Lets just say I have 30 apples and I want to consume them over a 20 day period.  I want to eat 2 apples a day for as many days as possible; and then cut back to one apple so that I am finishing the last apple on the 20th day."

We can see that you would eat two apples in ten days and one apple for the remaining ten. However, I played around with some numbers and did come up with a formula that will generate the answer to this question.

First, let's set some variables.

a = Total Number of Apples (or object of choice)
d = Total Number of Days (or time interval of choice)
n = Smaller Apple Portion (or object of choice)
x = Number of Days to eat bigger apple portion

To find n, all you have to do is divide d into a and ignore the decimal following. For your example, 30/20 = 1.5, so the smaller apple portion is equal to 1 apple.

To find x, you can use the following formula:

x = a - dn

For your example, you would do:

x = (30) - (20)(1)
x = 30 - 20
x = 10

Ten would be the number of days you eat two apples. To find out how many days you will eat a single apple, just subtract x from the total number of days.

So, if you wanted to eat 50 apples in eight days, you would do the same thing. 50/8 = 6.25, so six is the smaller interval.

x = (50) - (8)(6)
x = 50 - 48
x = 2

You would eat seven apples for two days and six for the rest.

I was really happy to see some feedback and ideas for posts, and I would love for all of you to contribute as well. This is the hundredth post, and I think everything was something that I chose to post. I would really like you guys to email me your favorite cool math thing, and I will be more than happy to post it.