Seven boys and five girls are seated around a circular table with 12 chairs. Prove that there are two boys sitting opposite one another.

Question

Seven boys and five girls are seated (in an equally spaced fashion) around a circular table with 12 chairs. Prove that there are two boys sitting opposite one another.

I used 'G' for girls and 'B' or boys. I drew 7 B's and 5 G's in a circle, alternating G's:

B - G - B - G - B - G - B - G - B - G - B - B

I understand this is a pigeonhole principle problem, but I'm not sure how to show that this is true.

You have the sets: {B,G} , {B,G} , {B,G} , {B,G} , {B,G} , {B,B}

After the hints given, this is my solution: Given a circular table with 12 chairs, there are 6 pairs of individuals siting opposite each other. There are only 5 girls, but there are 7 boys. Therefore, by the pigeonhole principle, one of the pair of individuals must be 2 boys.

Answer 1

HINT: There are $6$ pairs of individuals sitting opposite each other. There are $7$ boys.

Note that it does little good to look at a particular arrangement of boys and girls, since the question is about all possible arrangements.

Answer 2

Suppose not, then each boy has a girl sitting opposite him.