A few months ago, I was sitting around bored on my computer, when my dad forwarded me an email. The email contained a link to a mathematics puzzler. For those of you too afraid to click the link, or who simply don't want to, the trick goes as follows:

1) Have a volunteer write down a three or more digit number, with the stipulation that the number should contain at least 2 different kinds of digits (i.e. numbers like 333 are not allowed). Make sure they don't show you this or the following two numbers.

2) Now, have them create a second number by rearranging digits from the first number. (i.e. if their number was 8745, their second number could be 7854, or 4785).

3) Have them create a third number by subtracting the 2 numbers from each other (bigger from smaller generally, although this does not actually matter).

4) From this third number, have them circle one of the digits (but don't circle a zero, as those are already a circle ;-P). Now, have the volunteer tell you all of the digits except for the circled digit. Done correctly, you will be able to tell them what their circled digit is correctly, every time, in a matter of seconds.

How does this work? Read on for more!

Step 1: The Additive Digitial Root

   The ability to perform this trick relies heavily upon an obscure mathematical concept known as additive digital roots. Historically, digital roots were used as a type of checksum to catch errors from mental and pencil-and-paper math methods in the dark and forgotten age before calculators.
   The additive digital root of a number can be computed by adding all of the digits of a number together to produce a second number, then adding all the digits of that number together, and recursively adding all the digits of a given sum until only a single digit remains. This digit is the digital root of a given number. For more details about using and computing digital roots, check out my instructable on the topic.
   By plotting the digital root of a number, a more precise definition of the digital root can be obtained. The digital root is 0 if the number is 0, 9 when the number is 9, the number mod 9 when the number is not a multiple of 9, and 9 when the number is a multiple of 9. This, coupled with the fact that digital_root(x-y) = (x-y)mod9 will allow the math guru to read the mind of his unsuspecting volunteer.

Step 2: Applying the Digital Root

Let's review the "mind-reading" instructions, and see how the digital root applies to the technique. For review, the technique steps are listed below:

1) Have your volunteer pick a 3 or more digit number containing at least 2 different digits (i.e. numbers such as 333 are not allowed).

2) From the first number, tell them to create a second number by rearranging the digits of the first number (i.e. if your number was 321, your second number could be 132, 231, etc.)

3) Have them make a third number by subtracting the first and second numbers from each other (it doesn't matter the order of the subtraction, though as a personal preference I generally prefer a positive difference so I ask that the smaller number be subtracted from the larger one).

4) Now, have your volunteer one of the digits from the third number (but not a zero, as it's already a circle, and for reasons we will discuss shortly), and tell you all of the digits except for the circled one.

So how do you guess the number? We can reason this out in a series of steps:

1) We started by having the volunteer pick a 3+ digit number, but added that it needs to have at least 2 unique digits so that when they perform the subtraction they don't end up with 0 as a result (which, while it still technically works makes for a lousy trick). This number will have a given digital root, which we will call x.

2) When the volunteer rearranges his number, it will still have the same digital root x. This is because the order in which the digits are added does not affect the digital root sum of the number. Thus, number 2 also has a digital root of x.

3) When the subtraction is performed, the third number will have a digital root of x-x=0 (The difference of the digital roots of the previous 2 numbers), but we have ruled out 0 as a possibility for the difference, so the difference modulo 9 being 0 suggests the number is a multiple of 9, and therefore the number has a digital root of 9.

4) We had to prevent the volunteer from circling a zero, and the reason for doing so is that both 0 and 9 have no impact on the digital root of a number. For example, 10 and 19 both have the same digital root. The volunteer then reads off all of the digits except the uncircled digit. Since we know the number is a multiple of 9, and a 0 was not circled, simply choose the number that makes the sum of their uncircled digits 9 or a multiple thereof (Note that if the digits already add up to a multiple of 9 then the circled number must be 9).

Step 3: Examples

Let's try a few examples to illustrate the technique.
Example 1:

Suppose the volunteer chose the number 2331. The curious will note that 2331 has a digital root of 2+3+3+1. = 9
Now, suppose for a second number, the volunteer mixed the digits around and chose 1332. It should be noted this has a digital root of 1+3+3+2 = 9, the same digital root of the original number.
When they subtract the two numbers, they will get 2331 - 1332 = 999. The expected digital root of the difference is 9-9 = 0 (but in the setup we ruled out the possibility of a difference of 0, so the digital root must be 9). Running a check of the number, we see that the digital root of the difference is indeed 9+9+9 = 27 => 2+7 = 9, which is what we expect.
Now suppose the volunteer decides to circle a 9. They then tell you that their two numbers are 9 and 9. Quick thinking tells the operator that they must add 9 to get to the next multiple of 9, so the circled digit must be 9.

Example 2:

Impressed but unconvinced of your psychic abilities, the volunteer decides to try again, this time picking the number 4820. (4820 has a digital root of 5, for those keeping track)
For a second number, they pick 0248 (which also has a digital root of 5)
The difference of those 2 numbers is 4820-0248=4572 (the digital root of which is 9, for reasons previously discussed)
The volunteer tells you the numbers 4,5, and 2. 4+5+2 = 11. You deduce that their circled digit must be 7, as that is needed to make the sum a multiple of 9. (And you're right!)

Try it out! Amuse and mystify your friends, and, if you like it enough or have something to add, go ahead and rate or comment below.