   The sum of
the digits in
the product of
any number and
nine will always
sum to a multiple
of nine. The difference
of any number and
the sum of its
digits will always
result in a number
that is a multiple
of nine. The difference
between the larger
and smaller of two
numbers consisting
of exactly the same
digits will always
result in a number
whose digits sum
to a multiple of
nine. ## The First Mysterious Nine Trick

1. Have a friend think of a number between 1 and 9.
2. Then, ask them to multiply this number by 9.
more than one digit.
4. When they let you know they have finished, pretend you are thinking real
hard and then tell them they got nine.

If they have a pencil and paper handy, you can ask them to sum the digits in the
product of any number and nine, until they get one digit. Once they arrive at
that digit, it will always be nine.

Take a look at the nine times table below. Do you see why this works?
1 x 9 =   9..........9 = 9
2 x 9 =  18..... 1 + 8 = 9
3 x 9 =  27..... 2 + 7 = 9
4 x 9 =  36..... 3 + 6 = 9
5 x 9 =  45..... 4 + 5 = 9
6 x 9 =  54..... 5 + 4 = 9
7 x 9 =  63..... 6 + 3 = 9
8 x 9 =  72..... 7 + 2 = 9
9 x 9 =  81..... 9 + 1 = 9
10 x 9 =  90..... 9 + 0 = 9
11 x 9 =  99..... 9 + 9 = 18 and 1 + 8 = 9
12 x 9 = 108..... 1 + 0 + 8 = 9

Now, let's try this with a random number, say 1492.

1492 x 9 = 13,428
Now add the digits 1 + 3 + 4 + 2 + 8 = 18. Since18 is more than two digits,
keep adding across until you get one digit. 1 + 8 = 9!

Try it for yourself.

## The Second Mysterious Nine Trick

1. Have a friend pick a random three or four digit number. Yes, this trick
will work for digits of any length, but don't make it too difficult for
the other person.
3. When they have done this, ask them to subtract this sum from the number.
4. After they subtract, ask them to circle one of the digits in their
answer that is not a nine, and tell you only the numbers they did not circle.
5. To find the digit they circled you must add the digits they did not
circle, and then use the following rules:
• Rule number one: if the sum of the digits is less than or equal to 9,
subtract this sum from 9.
• Rule number two: if the sum of the digits is more than 9, add the
digits again until you arrive at single digit, and then subtract
this number from 9.

Your result will be the number they circled.

Here is an example:
Random number: 7832
Sum the digits: 7 + 8 + 3 + 2 = 20
Subtract this sum from the original number 7832 - 20 = 7812
Circle one of the digits in the answer.

Let's assume your friend circled the 2 in the number 7812. In this case the
other person would tell you that the numbers not circled were 7, 8, and 1.
You add these, and get 7 + 8 + 1 = 16. Since 16 is more than two digits you keep
adding digits to get 1 + 6 = 7. Since 7 is less than 9 follow Rule number one and
subtract this sum from 9. 9-7=2. 2 is the number they circled!

Here is why this works.
For any random number, if you subtract the sum of the random number's digits from
the original number, the sum of the digits in the difference always sums to 9.
Look at the example above.
The difference was 7812, and 7+ 8 + 1 + 2 = 18, and 1 + 8 = 9!

So, if you know that the sum of the digits in the difference is always 9, and
you know all the digits except one, then the number circled plus the ones you
know must sum to nine. What you know plus what you don't know = nine,
therefore, nine minus what you know is what you don't know. Got it?

Let the computer guess

## The Third Mysterious Nine Trick

1. Have a friend pick a random three or four digit number, for example 3428
2. Ask them to scramble the same digits to form a different number, for example 4823
3. Tell them to subtract the smaller from the larger, 4823-3428 = 1395
4. When they have done this, ask them to circle one of the digits in the
answer that is not a nine and tell you the ones not circled.

Let's say that the other person circled the 5, and reported that
the numbers not circled were 1, 3, and 9. To find the number they circled; add
the numbers they did not circle and use one of the following rules to
solve the mystery.

• Rule number one: if the sum of the digits is less than or equal to 9,
subtract this sum from 9.
• Rule number two: if the sum of the digits is more than 9, add the
digits again until you arrive at single digit, and then subtract
this number from 9.

Your result will be the number they circled.

In this case, the sum of the digits is 13. 1 + 3 + 9 = 13 so we would follow Rule
number two, and tell the other person that the number they circled was a 5
after first adding the 1 and 3 in 13 to get 4, and then subtracting this from 9.

Here is why this works.
For any random number, if you rearrange the digits in this number, and then
subtract the smaller from the larger, the sum of the digits in the difference
always sums to 9, if you keep summing until you get only one digit.

Using the example above, the difference was 1395, 1 + 3 + 9 + 5 = 18 and 1 + 8 = 9!
So, if you know that the sum of the digits is always 9, and you also know all
the digits except one, then the sum of the number circled and the ones you were
told must equal nine. What you know plus what you don't know = nine,
therefore, nine minus what you know is what you don't know. Got it?
Would you like the computer to guess?