The Mad Hatter, March Hare and Dormouse are sitting down for a tea party. They sit at a table with twelve chairs, and twelve cups of tea.

Each day at six o'clock, everyone moves over two seats to the left or to the right (if any of those seats are free), then if there is tea in the cup at their seat, each one drinks it so the cup becomes empty.

After this, Alice comes and fills one of the empty cups on the table with tea again.

Prove that Alice can make sure that there are at least six full teacups on the table every day just before six.