We investigate infinite variants of the following hat-colour guessing game. There are prisoners standing in a line, all facing towards the same end of the line. Each prisoner is given a hat which is either red or blue. Each prisoner can see those prisoners' hats who are in front of them. They guess the colour of their own hat in the order they are standing, starting with the prisoner who sees everyone else. They hear each other's guesses and can use this information to guess their own colour. The question is to determine the minimum number of incorrect guesses that can be guaranteed, and to give a strategy achieving it. For example, we will work on the cases when the prisoners stand on the rationals or .