As a follow up to my article “The Magic of One Numbers Part I” I now continue with Part II in this fascinating series. For those who have not read the first article, please do so now so that you can better understand this one. Here I will show you a method to perform the multiplication of any two “one” numbers regardless of size. The result of such multiplication—once this method is studied and learned—can be obtained effortlessly and usually within seconds. So let’s get started.

In the first of this series, I showed how to square any number which consisted of a series of 1's. Thus after learning this method, one could square 11 or 111. In this article, you will learn how to multiply two arbitrary “one” numbers together, such as 11 x 111. To do these multiplications, you need only learn a simple rule and the rest—well the rest—will be simply matter of fact. After thoroughly mastering these two techniques, you will be able to mesmerize people with your new-found math skill; and for those parents out there teaching these techniques to their kids, don’t be surprised if you get some phone calls from your kid’s math teachers, after your kid has demonstrated to them these powerful and novel methods.

This method is a little more involved than the squaring technique; however, with a little thought and practice, you will come to see that it really is no more difficult to master. Let us look at the example of multiplying 11 x 111. The result is 1221. The way we arrive at this result is by making some observations and then following a simple procedure. First we observe that the smaller “one” number, 11, has two 1's. Both numbers have a total of five 1's. The final answer will have a number of digits equal to 1 less than the total of 1's in both numbers, or in this case 4 digits. The answer, 1221, is obtained by noticing that if we count from 1 consecutively up to the number of 1's in the smaller “one” number and then down from that number without repeating it, we have 1 2 1, or only three digits. We need four in the answer so we insert another 2 between the 2 and 1 to get 1221.  This is always the case and the number we use to “pad” the answer, so to speak, is the number which represents the number of 1's in the smaller “one” number.

A few more examples should make this perfectly clear. Let’s look at 11 x 1,111. The total number of 1's in both numbers is 6. So the answer will have 5 digits. Since 2 is the number of 1's in the smaller “one” number, and if we count 1 2 1, we have only 3 digits; however, we need 2 more, so we pad the number with two more 2's in the middle to get 1 2 2 2 1 or 12,221 as our final answer.

Take 111 x 1,111. A total of 7 1's so our final answer will have 6 digits. Number of 1's in the smaller number: 3. So count 1 2 3 2 1 and observe that this consumes 5 digits. We need 6 so we pad 1 more 3 in the middle to get 1 2 3 3 2 1 or 123,321. To wrap up, I’ll show one more example and then you can go off amazing your friends and family. Take 1,111 x 111,111 or one thousand one hundred eleven times one hundred eleven thousand one hundred eleven. How many total 1's: 10. So the answer will have 9 digits. Number of 1's in the smaller number: 4. So we count up to 4 and back from 4 to get 1 2 3 4 3 2 1 and observe that this uses 7 digits. We need 2 more so we pad with 2 more 4's to get 1 2 3 4 4 4 3 2 1 or 123,444,321 or one hundred twenty-three million four hundred forty-four thousand three hundred twenty-one as our final answer.

What do you think now? Do you think that armed with these techniques your kids could get better math grades? I think that’s a rhetorical question. Good calculating. 

Joe Pagano