Two Dots is NP-complete

Two Dots is a popular single-player puzzle video game for iOS and Android. In its simplest form, the game consists of a board with dots of different colors, and a valid move consists of connecting a sequence of adjacent dots of the same color. We say that dots engaged in a move are "hit" by the player. After every move, the connected dots disappear, and the void is filled by new dots (the entire board shifts downwards and new dots appear on top). Typically the game provides a limited number of moves and varying goals (such as hitting a required number of dots of a particular color). We show that the perfect information version of the game (where the sequence of incoming dots is known) is NP-complete, even for fairly restricted goal types.