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.
Since they can only move two seats to the left or right, it follows that each participant must stay on either odd or even number seats at all times.
If the three characters have the same parity-then it follows that that all three cannot get to the cups of the other parity, and those six cups will always stay filled. Problem solved for Alice.
If the three charaters have different parity. This is possible iff one of the characters has a different parity from the other two. Then this will be the character who drinks the tea from the set of 6 cups. Alice simply needs to fill up the cup each time the character drinks it.
Puzzle Solution
