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/
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