## Thursday, October 15, 2009

### A Curious Number Trick

Here is a little number trick you can use to amaze your friends. The idea is to pick any number between 10 and 1,000, and then carry out the algorithm below. The resulting answer will always be 42 no matter what number you choose. Imagine that! As someone devoted to a life of science and mathematics you realize there is no magic here so please submit your proofs here on the Science Lifestyle Blog. See who can come up with the correct proof first.

Tell your friend to pick a number.

[She picks 575]

Add the digits together.



Add the digits again.



Add 3 to the result.



Subtract this result from the original number.



Add the digits together once again.



Find the remainder left when you divide the result by nine.



Square the result.



Now add 6 to the result.



The result is always 42! Try it again to see.

#### 2 comments:

1. Really easy to prove.
The trick is to drop unnecessary complexity.

After summing twice consecutive time the digits of the original number, we fall in [1;9] lets call each item A. (Remainder of division by 9)

Then we add 3 to fall in [4;12] lets call each item B.

If we do the substraction in Z/Z9, B-A is 6.

That's it !

2. Here is the same extended :

Tell your friend to pick a number.

{Let's call that number X}

Add the digits together.

{Let's call the result X'}

Add the digits again.

{Let's call the result X", X" is X mod 9}

Add 3 to the result.

{Let's call Y as X+3, Y is in [4,12]}

Subtract this result from the original number.

{So Z=X-Y (notice that X>Y)}

Add the digits together once again.

{We can drop this since we are going to find the remainder and that it is part of this process}

Find the remainder left when you divide the result by nine.
{Let's call it R as Z mod 9}

{We did the following (X-R) mod 9. (X>R). Let's call it R"}

Square the result.

{R" is always 6, we obtain 36}

Now add 6 to the result.

[36+6=42]

The result is always 42! Try it again to see.

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