You are trapped in a room. In front of you, there is a table on which there is an infinite number of stacks of cards each containing an infinite number of black or white cards.
Two stacks are considered to be "similar" if when you put the stacks side by side, every $p$-th pair of adjacent cards has the same color (for some prime number $p$).
You have a infinite number of empty tables, each labeled with a positive integer. To be able to leave the room, you must move stacks onto the other tables such that no table contains two similar stacks.
You look at the stacks, and see that for any stack, it would be possible to move all stacks that are similar to it onto the other tables. However, you're not sure how to move all stacks such that there's no two similar stacks on the same table.
Is it possible to achieve this? If yes, how?
Notes: A table can contain uncountably many stacks. You are able to move uncountable many stacks. You can use the axiom of choice. I'll be adding one hint every two days.