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/
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