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### 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).

some more, or if you have any other questions.

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

```
Associated Topics:
High School Discrete Mathematics
High School Puzzles
Middle School Puzzles
Middle School Word Problems

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