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Math Curiosities
• Can you add the numbers from one to 100?
• The curious thing about multiples of 9.
• How tall is a stack of a million one dollar bills?
• How to measure the thickness of a single page with a 12 inch ruler
• Playing with dice
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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.
3. Next, tell them to add the digits in their answer, if their answer had 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

Have a friend pick a random three or four digit number. This trick will work for numbers of any length, but don't make it too difficult for the other person.

Next, ask them to add the digits in the number. Lets say they picked 123. Then the sum of the digits would be 1+2+3 or 6

After they have added the digits in the number they selected, ask them to subtract this sum from the original number they picked. In our sample case this would be 123-6 which is 117.

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.

To find the digit they circled you must add the digits they did not circle, and then use the following rules:

1. If the sum of the digits is less than or equal to 9, subtract this sum from 9. If greater, use rule two.


2. 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: Let's say they pick a number like: 7832.

We ask them to sum the digits: 7 + 8 + 3 + 2 = 20

Next we have them subtract this sum from the original number 7832 - 20 = 7812

Finally we ask them to circle one digit in the result. 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.

To find the number they circled use rule number one and add the numbers they did not circle, 7 + 8 + 1 = 16. But Since 16 is more than one digit, we continue with rule number two and add again. 1 + 6 = 7.

Now that we have only one digit we subtract it from 9 and that is the number they circled. 9-7=2!

Here is why this works. For any random number, if you subtract the sum its digits from the original number, the digits in the difference always sums to 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 must equal nine, therefore, nine minus what you know is what you don't know. Got it?

The Third Mysterious Nine Trick

Have a friend pick a random three or four digit number, for example 3428. Ask them to scramble the same digits to form a different number, for example 4823. Next tell them to subtract the smaller number from the larger, 4823-3428 = 1395.

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.

To find the number they circled, follow the same rules as for the Second Nine Trick above. It will work for this case as well.

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, apply the two rules for nines and your result will be the number they circled.

In this example, the sum of the digits is 13. 1 + 3 + 9 = 13 so we would follow Rule number two, continue to ad until we arrive at 4, and tell the other person that the number they circled was a 5 since 9-4=5.

1. If the sum of the digits is less than or equal to 9, subtract this sum from 9. If greater, use rule two.


2. 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.

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?