Start with any number (but NOT 1)

Say 

Square that number:



Subtract 

 

Divide by one less than your original number.

The original number was .  One less than  is .
So dividing  by 



Now subtract your original number.



You reached .

-------------------------

That works for all numbers. Why does it work?  

Start with any number

Say 

Square that number:



Subtract 

 

Divide by one less than your original number.

The original number was .  One less than  is .
So dividing  by 



Now subtract your original number, which was 



You reached .

Notice that you could not have started with 1 because you
cannot divide by 1 less than 1 because that would be dividing
by zero!

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You can make up a rational expression with a factor in the bottom
that will cancel a factor in the top.  Say you start with this



Then multiply the top out:



Then to make the numerator , you say

Start with any number (other than 2).

Square it (that gives 

Add the original number (that gives 

Subtract 6  (that gives 

Divide by 2 less than the original number (that gives 

Subtract your original number  (that gives 

You reach 3.

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Try it with an arbitrary number.

Start with any number (other than 2).  Say we start with 83

Square it (that gives 

Add the original number (that gives 

Subtract 6  (that gives 

Divide by 2 less than the original number (2 less than 83 is 81.


Subtract your original number  

You reach 3.
 
You cannot start with the number 3 because you would have to
divide by 0.
 
Edwin