How Does the Trick Work?

Date: 06/20/2003 at 01:28:59
From: Ros
Subject: Mathematics Patterns/Tricks

Dr. Math, 

From time to time I've run across math tricks that ask me to give a 
number, then take me through a series of other questions and by the 
time I work my way to the end, the trick tells me my original number. 
How does this work?

Date: 06/20/2003 at 13:04:27
From: Doctor Peterson
Subject: Re: Mathematics Patterns/Tricks

Hi, Ros.

Here is an example of such a trick:

   Birthday Number Puzzle Variations
   http://mathforum.org/library/drmath/view/61702.html 

As you'll see there, you can see how the trick works by doing it not 
with a specific number, but with a variable. Then you find that all 
that is being done undoes itself in the end; the expression you are 
evaluating simplifies down to "x".

It wouldn't look like much of a trick if I said this:

  Pick a number
  Multiply it by 7
  Divide that by 7
  Ta-dah! You've got your number back!

It looks just a little more impressive if I say:

  Pick a number
  Multiply it by 7
  Add 35 to that
  Subtract 35 from that
  Divide that by 7
  Ta-dah! You've got your number back!

You can still see that you are just undoing each thing you did.

But it I tangle up this expression,

  (((x * 7) + 35) - 35) / 7

by instead doing this:

  (((x + 5) * 7) - 35) / 7

which means just the same thing, but doesn't have you both adding and 
subtracting 35, then it starts to look surprising:

  Pick a number
  Add 5 to it
  Multiply that by 7
  Subtract 35 from that
  Divide that by 7
  Ta-dah! You've got your number back!

If I did a little more to twist it up, you'd never guess the outcome!

All these tricks do is to hide the fact that you are undoing 
everything you did.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/