Bulgarian Goats

Date: 05/17/2003 at 05:55:37
From: Jane
Subject: Bulgarian Goats 

Yenko the Bulgarian goatherd drives his father's goats into a valley 
each morning and lets them browse there all day before driving them 
home in the evening. 

He notices that each morning the goats immediately separate into 
groups and begin to feed. The number and sizes of the initial groups 
vary. Some days there are nine or more groups; on other days, there 
are three or fewer. There can be groups of one or the whole herd can 
form a single group.

About every five minutes one goat breaks away from each feeding group 
and these breakaways form a new group

Yenko has noticed that by the afternoon, even though the goats 
continue their regrouping, the sizes of the groups have stabilised, 
and there are always seven feeding groups.

Q1. How many goats are there in the herd? What are the sizes of the 
feeding groups once they have stabilised? (The answer has to include 
the correct number of goats, the group sizes with justification, and 
the uniqueness of this set of group sizes.)

Yenko's father then sells two of the goats. Over the next week, Yenko 
notices that things have changed. The sizes of the feeding groups no 
longer stabilise. There are not always seven groups. Nevertheless, a 
cyclic pattern of sizes develops every day.

Q2. Find at least two possible cyclic patterns of sizes. (The answer 
has to include a correct pattern with appropriate evidence of 
investigation and another correct pattern with appropriate evidence 
of investigation.)

Date: 05/17/2003 at 09:49:26
From: Doctor Jacques
Subject: Re: Bulgarian Goats 

Hi Jane,

Let us look at what happens in a simple case. Assume we have three 
groups, of 2, 4, and 7 goats.

 * *
 * * * *
 * * * * * * *

One goat separates from each group:

 * | *
 * | * * *
 * | * * * * * *

and these (three) goats gather to create a new group:

 *
 * * *
 * * * * * *
 * * *

What happened is:

* Every group was reduced by 1

* A new group was formed: this group has as many goats as there were
  groups before.

Now we are told that, after some time, there are always 7 groups. In 
the example above, you noticed that a new group was formed. If the 
number of groups must stay the same, it means that one of the previous 
groups has disappeared, and the only way that can happen is if that 
group contained a single goat.

We see that we must have one group of one goat. Note also that the new 
group will contain 7 goats (one from each original group).

A little later, the process is repeated, and the number of groups 
still does not change. This means that one of the new groups now 
contains a single goat. As the groups have been reduced by 1, that 
group originally contained 2 goats.

Do you see where this leads? Note that we only consider what happens 
when the number of group has stabilized - the initial sizes may have 
been different.

For the second part of the question, you know the number of goats (2 
fewer than before), and you should just experiment with some initial 
configurations containing the correct total number of goats. You pick 
a configuration, and you repeat the process (for example, by using 
the kind of diagram I showed you) until you find a configuration that 
has happened before (it will not always be the first one); after 
that, the pattern will repeat. Note that, as there is only a finite 
number of possible configurations, you will always end up with a 
configuration that has occurred before.

For a simpler example, if we start with 2 groups of 1 and 4 goats, we 
have the following configurations:

a *
  * * * *

b * * *
  * *

c * *
  *
  * *

d *
  *
  * * *

e * *
  * * *

and you can see that we are back to the second state (b), with two 
groups of 2 and 3 (the order is not important).

Does this help?  Write back if you'd like to talk about this 
some more, or if you have any other questions.

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