KöMaL 2025 B5483

A deck of $2n$ cards consists of $n$ suits, each suit consisting of two values, 1 and 2. $n$ players play the following cooperative game: They start by shuffling the $2n$ cards, then each player is dealt two cards, which they reveal to the other players. Subsequently, they decide together the order they sit down around a round table, and they also choose the player starting the game. During the game, each consecutive player (in the clockwise direction) tries to put down a card, however, a card with value 2 can only be put down if the card of the same suit with value 1 is already on the table. They win the game if they manage to put all the cards on the table, however, they lose the game if somebody holding at least one card in their hand cannot put down a card on the table. Prove that the players can always win the game, regardless of the hand they were dealt.