  In this paper we introduce and investigate a new type of automata which turns
out to be rich in deep and complex phenomena. For our model, a maze is a
countable strongly connected digraph called the board together with a proper
colouring of its edges (the edges leaving a vertex have distinct colours) and
two special vertices: the origin and the destination. A pointer or robot starts
at the origin of a maze and moves naturally between its vertices, according to
a finite or infinite sequence of specific instructions from the set of all
colours called an algorithm; if the robot is at a vertex for which there is no
out-edge of the colour indicated by the instruction, it remains at that vertex
and proceeds to execute the next instruction in the sequence. The central
object of study is the existence of algorithms that simultaneously solve, that
is guide the robot to visit the destination in, certain large sets of mazes.
  One of the most natural and interesting sets of mazes arises from the square
lattice $\Z^2$ viewed as a graph with arbitrarily many edges removed (each edge
corresponds to a pair of opposite directed edges), together with the suggestive
colouring that assigns to each directed edge the corresponding cardinal
direction. In this set-up, a research question of Leader and Spink from 2011,
which proved to be very profound, asks whether there exists an algorithm which
solves this set of mazes.
  In this paper we make progress towards this question. We consider the subset
of all such mazes which have arbitrarily many horizontal edges removed but only
finitely many vertical edges removed in consecutive columns, and construct an
algorithm which solves this subset of mazes.
