# Swapping eggs puzzle

You can try this with pencil and paper, or make it a physical puzzle you can try out with your kids if you have one of those large 3x6 egg cartons laying around.

Imagine you have 3 rows of 6 spots. Fill the first row with 6 red colored eggs, the second with 6 green colored eggs, and the last with 6 blue colored eggs.

You are allowed to make as many moves as you like by following these steps:

1. Choose two rows and two columns.
2. At the intersection is four eggs: swap the top two eggs with the bottom two eggs.

First part of the puzzle:
Starting from this state

[RRRRRR]
[GGGGGG]
[BBBBBB]


can you reach this diagonally striped state?

[RGBRGB]
[GBRGBR]
[BRGBRG]


If not, prove why.

Second part:
Can you reach this state?

[RRRRRB]
[GGGGGG]
[BBBBBR]


If not, prove why.

I was able to solve the first part, and am stuck on the second part.
I assume it is some kind of parity argument, like most of these types of puzzles, but I cannot figure out how to demonstrate it.

I've filled a few sheets of paper doodling move sequences, and haven't spotted the "invariant" or pattern yet.
Help?

Basically, at every moment, each column corresponds to a permutation in $$S_3$$, the symmetric group of three elements.
The elements of $$S_3$$ are divided into two classes: the even permutations and the odd permutations. Any swap of two elements changes the parity.
Originally there are $$6$$ even permutations (all being identity) and the goal has $$5$$ even permutations. Hence it's impossible.