### Arithmetic without tables

When you look for patterns you may find amazing things.

2 * 6 = 12

4 * 6 = 24

6 * 6 = 36

8 * 6 = 48

Can you see a pattern?

The answer's second digit is the number we multiply six with and the first digit is half that number.

1 * 6 = 06

3 * 6 = 18

5 * 6 = 30

7 * 6 = 42

9 * 6 = 54

What is the pattern?

Let me take 7 * 6 as an example. Add 5 to 7 to get 12. 2 is the answer's second digit. Half of 7 rounded down is 3, plus the 'carry' from 12 is 4. That gives the answer's first digit.

These two patterns can be combined to a rule for multiplying any number by 6:

To each digit add half the neighbour to the right rounded down and 5 more if the digit is odd.

What is 749 * 6?

I will write this as 0749 * 6 and start from the right.

9 has no neighbour and is odd: 9 + 5 gives 4 with 1 carry.

4's neighbour is 9 and there is a carry: 4 + 4 + 1 gives 9 with no carry.

7's neighbour is 4 and is odd: 7 + 2 + 5 gives 4 with 1 carry.

0's neighbour is 7 and there is a carry: 3 + 1 gives 4.

The answer is 4494.

Study multiplication by 7 and come up with a similar rule.

These patterns were first discovered by Jakow Trachtenberg while he was a prisoner of war. He discovered that anyone who can take halfs rounded down and add small numbers can multiply.

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