# What makes this mnemonic device work for multiplication?

I stumbled across a mnemonic device related to multiplication, outlined on this wikipedia page. I see that it does work, but I'd like to know why. It works as follows (from the wikipedia page):

         →                 →
1 2 3             2   4
↑  4 5 6 ↓         ↑       ↓
7 8 9             6   8
←                 ←
0                 0
Fig. 1             Fig. 2


Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. For example, to memorize all the multiples of 7:

1. Look at the 7 in the first picture and follow the arrow.
2. The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
3. The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
4. After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
5. Proceed in the same way until the last number, 3, which corresponds to 63.
6. Next, use the 0 at the bottom. It corresponds to 70.
7. Then, start again with the 7. This time it will correspond to 77.
8. Continue like this.

These patterns can be used to memorize the multiples of any number from 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0 (0 is external and so the arrows have no effect on 0, otherwise 0 is used as a link to create a perpetual cycle).

In the interest of further clarification: you choose a figure to use by looking for your number in a corner position. If, for example, you wanted the multiples of 8:

1. Start at the bottom right corner of Fig 2, and follow the arrow left to the 6
2. Thus, the next multiple of 8 is the next number that ends in '6', so 16.
3. Then move to the rightmost position of the upper row and you see that your next multiple end in a '4', so it's going to be 24. Etc.

I imagine similar devices could be constructed for numeric systems with bases other than ten, and I bet it'd be interesting to understand what's going on here. Does anyone have any insight into this? Is this something that falls under the umbrella of Number Theory?

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Is Fig. 2 used anywhere? –  Trevor Wilson Feb 20 '13 at 22:41
@TrevorWilson - yes, it is. I added another example to explain it further. Hope that helps –  ivan Feb 20 '13 at 22:56

For $7$, note that the number above is always $3$ less than the number below. When you add $7$ to a number, you can also add $10-3$. For $9$, the number to the left is $1$ less and you are adding $10-1$. The evens work similarly.