Students in a class, girls sitting with boys and boys sitting with girls

Question

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately I was confused by the problem and I don't really understand it either, hence I am unable to show my own working or opinion. The textbook gave no hints really and I'm really not sure about how to approach it. Any guidance hints or help would be truly greatly appreciated. Thanks in advance :) So anyway, here the problem goes:

There are $30$ students in a class. They sit at $15$ double desks; each desk seats two students. Half of the girls sit with the boys. Is it possible to make a rearrangement so that half of the boys sit with the girls?

Answer 1

Regardless of the relative amounts, the sum of half the boys and half the girls gives half the total number of students. So that is $15$ students, who together need to occupy some number of $2$-seat desks. This is impossible for parity reasons.

Answer 2

It cannot be done because to have half of the girls sitting in pairs, so half of the girls must be an even number so the number of girls must be a multiple of 4.

The same argument applies to the situation in which half of the boys are sitting with girls, so the number of boys must be a multiple of 4.

But they can't both be multiples of four since they add to 30 which is not a multiple of 4.

Answer 3

Let there be $2G$ girls and $2B$ boys, $2G+2B=30$, so $B = 15-G$.

After seating half the girls with boys, we have G girls and $(2B-G) = (30-3G)$ boys.

For them to be equal, $G = 30 - 3G$, which yields $G = 7.5$, i.e. $15$ girls and $15$ boys at start.

I think that you will agree that it is rather difficult to seat two groups of $7.5$ girls and $7.5$ boys!