1. ## Permutations

25 boys and 25 girls sit around a table. Is it always possible to find a person both of whose neighbours are girls?
I don't think this is true. I worked with 5 boys and 5 girls to manufacture situations where it wasn't true. I want to avoid writing out a counterexample with 50 people!

Is there a way to approach this using permutations? It seems to be that if you put them in Boy/Girl order, and then switch the two people before the person with two girls as neighbours then you get a nice counterexample.

Swapping people around seems like it can be done using permutations.

Anyway, I hope someone can help.

2. Can you manufacture a situation for the 5 boy, 5 girl case? Because I can't.

You could argue that it if it holds for 25 boys and 25 girls, than it holds for two equal and odd amounts of boys and girls. Of course 1 boy and 1 girl does not work though.

Here is the 3 case. Boys=1 Girls=0

0 0
111

Now try to place the last girl.

3. Originally Posted by Anonymous1
Can you manufacture a situation for the 5 boy, 5 girl case? Because I can't.

You could argue that it if it holds for 25 boys and 25 girls, than it holds for two equal and odd amounts of boys and girls. Of course 1 boy and 1 girl does not work though.

Here is the 3 case. Boys=1 Girls=0

0 0
111

Now try to place the last girl.
This. I couldn't find a way to do 5 either, or for any odd number. For even it's easy, just have everybody arranged in 2,2,2,2, etc. But with odd you will have one extra of each boy and girl to slot in somewhere. Then wherever they go there will be someone with 2 girls on each side.

Post your solution to the 5 of each version. If you can do that then you can generalize for any odd numbered version.

4. Originally Posted by Showcase_22
25 boys and 25 girls sit around a table. Is it always possible to find a person both of whose neighbours are girls?
True.