## 7/26/2010

### A Magic Trick :: Guessing number

I saw one of student in English class taught us about how to do a magic trick to guess a number from other people.
He used 5 sheets of numbers on it, and he asked a volunteer to see the sheets one-by-one and tell him whether the number of the volunteer's birth date is in that sheet or not?
After the end of the 5th sheet, he shouted a number, and surprisingly it is the correct answer!!

How can it be possible??

First, i have to introduce a binary-number system. It is the same numerical system that has been using in computer system. As we can easily recognize because it has only 0 and 1.
Any decimal number can convert into a binary number. for example: 13 in decimal is 1101 in binary.
or 28 in decimal is 11100 in binary.
And how to convert it back? we sum up the value for each digit position that has '1' on it.
for example of 5 digits binary, the value for each position from left to right are
position:value for the position
1: 16
2: 8
3: 4
4: 2
5: 1
Example, we want to convert 11100 back to decimal, we can start by checking the position that has '1' on, which are position 1,2,3
then sum up the value for the position 1,2,3; we'll get 16+8+4 = 28

What is this related to the trick?

First, the sheets of numbers:
since the trick is created for guessing the number for birth date, maximum number is 31.
and if we convert 31 into binary, it is 11111 which has 5 digits. (sounds familiar?)
Because each sheet, we can think of it as position of the digit. One sheet represents one position.
and each number on each is the number that when it converted to binary, it has '1' on that position(sheet)
example
number 5 : convert to binary = 00101 . So, number '5' will appear in 2 sheets; sheet for position 3 and sheet for position 5.
number 13: ---> 01101 . then the number '13' will show on sheet for position 2,3 and 5.

Then, How can we guess the number?

Think of it as a yes/no question. '0' means 'No' and '1' means 'Yes'. Then when we show the sheets to the volunteer and we ask whether number he/she thinks of is in those sheets or not, we get "Yes"/"No" answer back. and that means we get series of '0' and '1' back
Then to reconstruct the answer, you just need to convert the binary number into decimal number
by summing up number for the "yes" answer that i stated before.

(From the picture attached) You can simply sum up the top-left number of the sheets that volunteer says "Yes", and that's the answer.
And It's just 1,2,4,8,16.... binary number.

That's it!

Sheets of numbers (image from: http://illuminations.nctm.org/LessonDetail.aspx?id=L245)