## 'Cows and Sheep' printed from http://nrich.maths.org/

Before looking at some solutions that came in we would like to say that we were very pleased with the explanations that came in with many of the solutions - exactly what we would hope to encourage.

Francesca and Alex sent in the first correct solution and they come from Red Hill Field Primary School and so does Sam who sent in another correct solution a week later.

Francesca and Alex wrote:

Here is the solution:
In field no.1 there must be 3 cows and 4 sheep because cows can't see themselves.
In field no.2 there must be 2 cows and 3 sheep.
In field no.3 there must be 4 cows and 6 sheep.
In field no. 4 there must be 5 cows and 8 sheep.
In field no. 5 there must be 4 cows and 9 sheep.
I know this because the animals cannot see themselves.

We had many good replies from Bradon Forest School and here's Samuel's correct solution with some good explanations as to how he went about it.

I worked out all of these questions by doing them step by step...
1) Step 1: Firstly, you must take 1 cow and 2 sheep. This is because each cow can see twice as many sheep than cows.
Step 2: It also says that each sheep can see the same number of sheep as cows. This means that there must be a higher amount of animals. Therefore there are 3 cows and 4 sheep . This is because each cow can see 2 cows and 4 sheep (remember the cow can't see itself) which is twice the amount of cows. and each sheep can see 3 cows and 3 sheep which is the same amount.

2) Step 1: First we take 3 times as many sheep as cows ( which is 3 sheep and 1 cow).
Step 2: Next, it tells us that each sheep can see the same number of cows and sheep. So therefore, we have to add one more cow. This means that there are 2 cows and 3 sheep. This is because each cow can now see one cow and 3 sheep and each sheep can see 2 sheep and 2 cows.

3) Step 1: Firstly, it says that each cow can see twice as many sheep as cows. This means there are 2cows and 2 sheep.
Step 2: It then says that each sheep can see one more sheep than cows, this means there are 6 sheep and 4 cows. This is because each cow can see 3 cows and 6 sheep. This is twice the amount of cows. It also means that each sheep can see 5 sheep and 4 cows. There is one more sheep than cows.

4) Step 1: We are given the information that each cow can see twice as many sheep than cows. This means there is 1 cow and 2 sheep.
Step 2: We are told that each sheep can see two more sheep than cows. Therefore, there are 5 cows and 8 sheep. This means that each cow can see 4 cows and 8 sheep which is twice as many sheep as cows and each sheep can see two more sheep than cows ( 1 sheep can see 7 sheep and 5 cows).

5) Step 1: We are told that each cow can see three times as many sheep than cows. So let's start with 1 cow and 3 sheep.
Step 2: It says that each sheep can see twice as many sheep as cows. This means there are 4 cows and 9 sheep. This is because 3 cows can see 9 sheep (3x3=9). Also each sheep can see 8 sheep and 4 cows which is twice the amount of cows.

Rees, who also came from Bradon Forest sent in a Powerpoint presentation and here's his slide for field 1

Well done!

We had 71 replies come in about this problem, They came from far afield - Qatar, Dublin, North Carolina, Australia, New York, Wisconsin and New Zealand as well as the U.K. The problem caused some deep thinking and just 16 replies had the correct answer.