The Crew: The Quest for Planet Nine is NP-Complete

In this paper we study the cooperative card game, The Crew: The Quest for Planet Nine from the viewpoint of algorithmic combinatorial game theory. The Crew: The Quest for Planet Nine, is a game based off of traditional trick taking card games, like bridge or hearts. In The Crew, players are dealt a hand of cards, with cards being from one of $c$ colors and having a value between 1 to $n$. Players also draft objectives, which correspond to a card in the current game that they must collect in order to win. Players then take turns each playing one card in a trick, with the player who played the highest value card taking the trick and all cards played in it. If all players complete all of their objectives, the players win. The game also forces players to not talk about the cards in their hand, and has a number of "Task Tokens" which can modify the rules slightly. In this work, we introduce and formally define a perfect-information model of this problem, and show that the general unbounded version, as well as the constant player count version are both intractable. However, we also show that two bounded versions of this decision problem -- deciding whether or not all players can complete their objectives -- can be solved in polynomial time.