The Farmer's Sheep

Problem

A farmer can see nine sheep if he looks out of any of his four windows. His wife buys him a new sheep. Which paddocks can he put the new sheep in so that he can still see nine sheep from each of the four windows?

In how many ways can the farmer put his 25 sheep so that he can see 9 sheep through each window? (Every paddock is to have at least one sheep.)

Teaching Sequence

1. Tell the students the story of a farmer who only likes to be able to see nine sheep from his window. His wife has bought him another sheep what is he going to do? Where can he put it?
2. Put a large version of the grid up in the classroom and point out the features e.g. where the windows are.
3. Explain the rules:
   The new sheep must go in a paddock, it cannot be destroyed, hidden, eaten, etc.
   The farmer must be able to see a total of nine sheep from the three paddocks he can see from each window.
4. Encourage the students to work in pairs and record the problem solving sequences they go through.
5. Share solutions. Is there a rule?

Other Contexts for the Problem

First of all the farmer can look at any animal he pleases or any crop he pleases (his wife might want to give him another pine tree). Second, you can think of the house as being a security camera. The security guard is happy if he can see 9 pieces of art on his surveillance cameras. Can you smuggle in a piece of art without the security guard being aware that anything has changed?

Solution

Perhaps the simplest way to do this problem is to notice that you can move one sheep from the top right hand paddock to the left centre one. This keeps 9 sheep in view from each window except the one at the top. If you put one sheep in the centre top paddock this balances up the problem.

But, there are heaps of other ways of doing this problem. However, they all need a lot more moving of sheep. Let the students come up with other solutions. At first this will probably be done on a guess and check basis. This may take a while. Perhaps they could do it as homework and get their parents or caregivers to help them. Make it into a competition. When you have a few more answers, you might begin to see a pattern. Surprisingly, in every case, the numbers of sheep in the four corner paddocks always adds up to 11. Why is this?

The following argument may be a bit hard but some of the more able students will be able to see what is happening. Suppose that we added together all of the sheep that can be seen from the four windows. Then we would come up with a sum of 36 (= 4 x 9). But if we add up all those sheep we are adding up the corner paddocks twice and the other paddocks once. So we are adding together every paddock once and the corner paddocks once more. ‘But every paddock once’ gives us all the sheep there are. This is 25. So 25 + the sheep in the corner paddocks = 36. This means that the corner paddocks have to have 11 sheep in them. Now all you have to do is to be systematic and you’ll get all the 16 answers below.