Tuesday, March 25, 2014

Number Magic

There are many tricks involving numbers that demonstrate their surprising relations to each other. What does any of this have to do with history? Well, number magic is as old as man's discovery of numbers.

An old chestnut is the way that the results in the multiplication table for the number nine run consecutively with each operation, but forwards in the tens column and backwards in the ones.

 1 x 9 =  9

 2 x 9 = 18

 3 x 9 = 27

 4 x 9 = 36

 5 x 9 = 45

 6 x 9 = 54

 7 x 9 = 63

 8 x 9 = 72

 9 x 9 = 81

10 x 9 = 90

This pattern actually continues after both columns start over at "9."

11x 9 = 99

Then the first two digits can be read together as consecutive numbers, counting up, while the last digit, in the ones column, counts down.

12 x 9 = 108

13 x 9 = 117

14 x 9 = 126

You can also see the results over one hundred as a "subtract one formula." For example, 4 x 9 = 36, so 14 x 9 can be solved by taking 36 and subtracting 1 from the 3, which equals 2, but keep the 1 before the 2, which gives you 1, 2, 6 and you have 126.

Also notice that if you consider the digits as separate numbers and add them up, you always get nine.

For example, 1 + 2 + 6 = 9, or take the sequential results 18, 27, 36 and 45: 1 + 8 = 9; 2 + 7 = 9; 3 + 6 = 9; 4 + 5 = 9. Also, consider 12 x 9 = 108. If you add 1 + 0 + 8, you get 9. Same with 13 x 9 = 117: 1 + 1 + 7.

Of course, one of the most obvious reasons why the first relationship works the way it does is that in a system of counting that is based on the number ten, we can easily see that adding ten to any number will change the number in the tens column one up, but the number in the ones column is the same; so 16 + 10 = 26, 17 + 10 = 27, and so on. Nine, however, is one less than ten; so when you add nine to any number, your ones column will always be one less than the number in the ones column of the number you were adding to. So, 9 + 10 = 19, but 9 + 9 = 18, or one less than 19. Each subsequent time that you add nine, the next result in the sequence is going to be one less in the ones column than the previous result.

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Here is another trick: The addition of any sequence of numbers where the last number is divisible by three, will always yield a sum that can be reduced to the number 6 if we add up its digits.

Examples:

Let's take 1, 2, 3.

1 + 2 + 3 = 6.

No need to do anything more in this case because the last number in the sequence is 3, which is obviously divisible by three (3 ÷ 3 = 1), and since six is the number we are looking for, we need look no further than the sum.

The next sequence (4, 5, 6), however, will give us a two digit sum, so we will have to add the component digits of the sum:

4 + 5 + 6 = 15.

And when we add the digits of this sum (1 + 5), we get 6.

This trick seems to go on working forever.

7 + 8 + 9 = 24; 2 + 4 = 6 ...

16 + 17 + 18 = 51; 5 + 1 = 6 ...

202 + 203 + 204 = 609; 6 + 0 + 9 = 15; 1 + 5 = 6,

and so on.

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