  We introduce a 2-player game played on an infinite grid, initially empty,
where each player in turn chooses a vertex and colours it. The first player
aims to create some pattern from a target set, while the second player aims to
prevent it.
  We study the problem of deciding which player wins, and prove that it is
undecidable. We also consider a variant where the turn order is not alternating
but given by a balanced word, and we characterise the decidable and undecidable
cases.
