Arithmetic Tricks - Nines

There is a nice trick you can do. Ask someone to get a calculator and multiply together a bunch of single digit numbers. Get them to go crazy, just multiply any old thing - as long as the result is displayed without rounding!

They then tell you all but one of the digits displayed on the calculator, and you tell them the remaining number.

This works because sooner or later they use a nine (or two threes, two sixes or a six and a three). When they do, all the digits will add up to nine. When they tell you the other digits, you just need to keep add them.

For example, suppose they have the total 34836480, these digits sum to 36, a multiple of 9. So, if they give you digits that sum to 33 (three less than a multiple of 9), you can then tell them that the missing digit was 3.

Small Sidenote

The reason the sixes and threes are equivalent is that 3 × 3 = 9 and 6 = 3 × 2, so if you have a six and a three, you are multiplying by 3 × 3 × 2, or 9 × 2, or 18 - a multiple of 9.

There is a chance that they don’t include these combinations, but if they multiply enough digits, that is unlikely - or malicious.

That the digital sum when multiplying by nine is a multiple of nine is a useful cross check when doing calculations.

Why does this work?

When you add the digits of a large number, you are performing a digital sum, and the digital sum of any not zero number is not zero.

Any large number can be expressed as follows:

243 = 2 × 100 + 4 × 10 + 3 × 1

Suppose we have a large number abcdef (where a to f represent digits) this number can be represented as:

f + 10e + 100d + 1000c + 10000b + 100000a

or

(f + e + d + c + b + a) + 9e + 99d + 999c + 9999b + 99999a

or

(f + e + d + c + b + a) + 9 (e + 11d + 111c + 1111b + 11111a)

or

(digital sum) + 9 × some number

Thus, if the digital sum is a multiple of 9, then the whole thing is a multiple of 9.

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