Sheep Shearing Rule

Date: 03/20/2003 at 23:42:13
From: Mitch
Subject: Shearing sheep

Eric the sheep is waiting in line to be shorn. Each time a sheep at 
the front of the line gets shorn, Eric sneaks up line four places. 

Describe a rule you could use to find the number of sheep shorn 
before Eric for any number of sheep in front of him.

Date: 03/21/2003 at 11:25:36
From: Doctor Ian
Subject: Re: Shearing sheep

Hi Mitch,

Interesting problem! Have you tried simulating it to see what happens?  
That's often a good way to get a feel for what's happening, which is 
the first step in finding a rule.

We start with some number of sheep in front of Eric:

  S S S ... S S S S S S S E 
  \_____________________/
         n sheep

The use of '...' and a variable for the number of sheep is a 
convenient way to avoid having to choose a particular number of sheep. 

Now, one gets shorn, 

  S S S ... S S S S S S E 
  \___________________/
         (n-1) sheep

and Eric sneaks up four places:

  S S S ... S S E S S S S  
  \___________________/
         (n-1) sheep

But we're only interested in the sheep in front of Eric, so we have

  S S S ... S S E S  
  \___________/
   (n-5) sheep

So if there were n sheep in front of him, there are (n-5) after the
first sheep is shorn. After the second sheep is shorn, there will be 

  (n-5) - 5

sheep, and after the third sheep is shorn, there will be

  ((n-5) - 5) - 5

sheep, and so on.

Is this enough to get started? 

Note that there is another way to think about it, which is this: From 
Eric's point of view, there's no difference between a sheep that is 
shorn ahead of him, and a sheep that moves to a place in line behind 
him. So another way to phrase the problem is this:

  Eric the sheep is waiting in line to be shorn. Each time a sheep 
  at the front of the line gets shorn, four more get shorn at
  the same time (in a single 'turn'). How many turns will it 
  take Eric to get to the front of the line? 

The number of 'turns' in this problem is the same as the number of
sheep shorn in the original problem. Does that make sense? 

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