In order to determine the missing digit, the mathemagician simply adds up the reported digits, and if that sum is more than 9, adds up the digits of the sum, until getting a single digit d. The missing digit is simply equal to 9-d. For instance, my reported digits were 899, which add to 26, which add to 8, so my missing digit was 1.
Why does this work? Well, the simple answer is that the result of the subtraction operation must be divisible by 9. Numbers that are divisible by 9 have the property that their digits add up to 9 (as long as you keep adding until you get a single digit). Now, that means that if you hide one digit, and report the others, even if they're scrambled, then all I have to do is add up the reported ones and subtract from 9. This procedure can't tell the difference between 9 and 0, however, which is why we have to forbid people from hiding a 0.
To see why the subtraction must result in a number divisible by 9, it's enough to think about the fact that we are using a base 10 system. Now think about the subtract of 3435 from 5433 as a set of four subtractions:
5433 = 5000 0400 0030 0003
-3435 = -0005 -0400 -3000 -0030
---- ----- ----- ----- -----
1998 = 4995 + ( 0000 ) + ( -2970 ) + ( -0027 )
Notice that every result on the right is divisible by 9. This is no
accident. But now we can see why. We arranged the subtractions
on the right so that digits match, and no matter what numbers are
involved in the subtraction, we can always do this as long as one number
is simply the scrambled version of the other. But now if we look at any
one of the simpler subtractions on the right, we see that, whether we
are subtracting 30 from 3 or 3000 from 30, we can always write this in
a form that reveals a 9:
30 - 3000 = 30 * ( 1 - 100 )
= 30 * ( -99 )
= -30 * 11 * 9
So if every term on the right is divisible by 9, then so is their sum,
which is the result of the subtraction. And that's why the missing digit
can be found.
Back to The Missing Digit Puzzle.