Give the following instructions to a friend:




1. Write down any number with 4 or more digits.

2. Rearrange the digits.

3. Subtract the smaller number from the larger.

4. Circle any of the digits (except a zero because it is already a circle).

5. Now have your friend give you all of the digits except the one circled (IN ANY ORDER).

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You will now tell your friend the digit that he or she circled.
(1) Find the sum of all the digits that your friend gives you.
(2) If the sum is less than 9, 
    subtract the sum from 9 and that is the digit circled.
(3) If the sum is 9, then your friend circled a 9.
(4) If the sum is more than 9, then add the digits of the sum together.
    If the sum is still greater than 9, 
    keep adding the digits together until you get a single digit.  
    Then subtract from 9 to get the digit circled. 

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Why does this work?




In step three, when you subtract the numbers, you always get a multiple of 9.
  That means that the sum of the digits is a multiple of 9.