The Secrets of Multiplication

Introduction (a Cute Story)

When I studied multiplication in third grade, a friend walked up and asked if I knew a fast way to produce the nine's times table. I said that I did not, so he produced a table in just four seconds. How he did it, really amazed me.

He took a sheet of paper and wrote a column of numbers from the top to the bottom starting at zero and finishing at eight. Then he started at the bottom and just to the right of the first column and wrote the numbers from one to nine going straight up. When I took a look at the resulting list of numbers, I was looking at the nine times table, completed in only four seconds!

The joy of the discovery only lasted for a moment. Then I asked an important question. Could this method, or something like it, be used for the other numbers? He looked at me as if I were stupid or something. My friend was not a mathematician. He just did not know any other method. He could not understand that it was even important to have anything beyond what he knew already. He walked away, disappointed that I was not impressed with his feat any longer than I was.

I however, had to try and find a way to do this very trick for all the rest of the numbers three through eight. The numbers one, two, and five were easily developed. The harder numbers were three, four, six, seven, and eight. These had no easy way to count out, so I began my search.

First Principles

I found a book soon after the incident above, and it contained algorithms that would generate the times tables for each of the numbers, but the methods were too complicated to be used by someone in the third grade. Generating the seven's times table was the absolute worst. I understood the math, but I sure could not do it in my head! It was better to just memorize the times table.

But I noticed that the numbers in the units place for one column were duplicated by the units place in another column for all of the numbers, only in reverse. The nine times table units were duplicated by the one's times table, only in reverse. The same went for the two's and the eight's, for the three's and the seven's, and for the four's and the six's. Also, the numbers in each of the paired columns would add to ten. This all led to a new generation method that could complete a times table column in just a few seconds. An entire times table could be developed in just a couple of minutes.

I had no ability to memorize stuff at the time to speak of, and this really made my test taking easy. Of course, I eventually memorized all of the times tables, but I have never forgotten what got me there.

The Method

To show my friends how I do it, I would draw a tic-tac-toe board and fill it in with the number one to nine from left to right starting at the top left. I use this as a generator for the odd numbers. By turning the board and putting the odd number that I wish to generate the times table for, in the upper left hand corner, the units can be read left to right. There are nine numbers to be copied. They are already in the proper order.










Let's say that you wanted to generate the seven times table quickly. First, you would turn the 3x3 table so that the number 7 is located in the upper left hand corner. You can do it on the table above. Just press the button until you get there.

Now you just read the numbers off in the order they appear from left to right. The ten's digits can be identified by simple logic. If the unit's number increases in the next number, then it will be the same as the one before it. If it decreases, then the ten's digit will have to increase by one. By this method, the ten's digits can be finished very quickly thus finishing the times table for that number.

When you read the numbers from the square, you get the following:

7, 4, 1, 8, 5, 2, 9, 6, 3.<===>7, 14, 21, 28, 35, 42, 49, 56, 63.

It is pretty much the same for the even numbers, but instead of using a 3x3 square, I use a 2x2 one. That is because I only have four even numbers to deal with.





To generate the even numbers for the even units, you turn the 2x2 square until the even number you need the times table of to the top left hand corner just like the odd number 3x3 square. Then you copy the four numbers left to right, top line first and then the bottom line. These are the first four units digits you need. The next units number is always zero, and then the first four repeat again.

For instance, say you wanted the six times table. You would turn the 2x2 table until the number six was in the top left corner. Then you copy the four digits in the order that they appear. You can check this on the square above. They are 6, 2, 8, 4. Then the next digit is always zero, and then the four repeat in the exact order. That's all there is to it!

The list of numbers: 6, 2, 8, 4, 0, 6, 2, 8, 4
With tens: 6, 12, 18, 24, 30, 36, 42, 48, 54.

With this kind of speed in generating number lists, a complete times table can be generated "out of the blue" in a minute or two by anyone who can not remember what the answers are. I hope you found this to be entertaining. Now, you can do more than just generating the nine times table. You can do all of them just as fast.