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Numerical Reverser

Part 2: Four Digit Reversal

If you haven't already done the Two-Digit Reverser, you may want to see it and read the description first.

You are asked to pick a four digit number. This is the equivalent to the following statement:

1000w + 100x + 10y + z

Where w, x, y, and z are all positive integers between 0 and 9.

The next step is to reverse the digits, which yields

1000z + 100y + 10x + w

And then to add the original number:

 1000z + 100y + 10x + w + 1000w + 100x + 10y + z
=1001w + 110x + 110y + 1001z

It may not be very obvious, but this expression is divisible by eleven! And, there is a useful property of multiples of eleven: the alternating sum of the digits is zero.

What does this mean? It means that if you add the first digit of the number, subtract the second, add the third, subtract the fourth, etc. until no digits remain, you are left with 0. And, since you're given the first four digits, it is simple to find the fifth. Here's an example:

Number: 1425?
1 - 4 + 2 - 5 + ? = 0
-6 + ? = 0
? = 6

Number: 0532?
0 - 5 + 3 - 2 + ? = 0
-4 + ? = 0
? = 4

It really is that simple. The only concern with this trick, of course, is doing four-digit addition in your head!