The Missing Digit

Tell your friend to write any multidigit number, but no ending in noughts, say, 847. Ask him to add up these three digits and then subtract the total from the original.

The result will be:
847 – 19 = 828

Ask him to cross out any one of the three digits and tell you the remaining ones. Then you tell him the digit he has crossed out, although you know neither the original nor what your friend has done with it.

How is this explained?

Very simply:

All you have to do is to find the digit which, added to the two you know, will form the nearest number divisible by 9. For instance, if in the number 828 he crosses out the first digit (8) and tells you the other two (2 and 8), you add the and get 10. The nearest number divisible by 9 is 18. The missing number is consequently 8.

How is that?

No matter what the number is, if you subtract from it the total number of its digits, the balance will always be divisible by 9. Algebraically we can take a for the number of hundreds, b for the number of tens and c for the number of units. The total number of units is therefore:

100a + 10b + c

From this number we subtract the sum total of its digits a + b + c and we obtain:

100a + 10b + c – (a + b + c) = 99a + 9b = 9 (11a + b)

But 9(11a+b) is, of course, divisible by 9. Therefore, when we subtract from a number by the sum total of its digits, the balance is always divisible by 9.

It may happen that the sum of the digits you are told is divisible by 9 (for example, 4 and 5). That shows that the digit your friend has crossed out is either 0 or 9, and in that case you have to say that the missing digit is either 0 or 9.

Here is another version of the same trick:

Instead of subtracting from the original number the sum total of its digits, ask your friend to subtract the same number only transposed in any way he wishes.
For instance, if he writes 8,247, he can subtract 2,748 (if the number transposed is greater that the original, subtract the original).

The rest is done as described above
8,247 – 2,748 = 5,499

If the crossed-out digit is 4, then knowing the other three (5, 9, and 9), you add them up and get 23. The nearest number divisible by 9 is 27.
Therefore, the missing digit is 27 – 23 = 4.