Is it possible to uniquely number faces of a hexagonal grid with consecutive numbers? You have a grid of regular hexagons.
The aim of the game is to have each hex contain the numbers 1-6 on its edges.
Each edge must also be connected to another edge that has a value one higher and one lower than the value of itself, with 6's wrapping to 1's. The other two edges can be any number as long as the first rule isn't broken.
Below, I have completed a small section by hand as an example:

Can this pattern propagate to a hex grid of infinite size and does it repeat?
In this example I started with a hex labeled 1-6 going anti-clockwise from the top however this is not a requirement.

This isn't so much a geometric problem about a hexagonal grid, this is very much a topological problem, specially a graph theory problem, involving an (infinite) graph, where the nodes are the vertices of the hexagonal grid and the edges are the edges. 
The problem you present reminds me very much of edge colouring (http://en.wikipedia.org/wiki/Edge_coloring), except whereas colours have no particular order or structure, your "colours" (i.e. numbers) come from Z_6 ( i.e. the integers modulo 6 , i.e.  the quotient group Z/6Z , i.e. {0, 1, 2, 3, 4, 5} with 5+1=0 ), and there are added restrictions of the "colouring" based on this group structure.
In fact, your problem isn't even restricted to tesselations in the plane. 
It can be abstracted to all "generalised polyhedra", which also includes such "tilings of polygons" as Euclidean polyhedra and hyperbolic tesselations. For instance, you could also ask the same question about the tiling of triangles on an icosahedron, or a tiling of pentagons where 4 pentagons meet at each vertex, even though such a regular tiling is impossible in the Euclidean plane. 
(http://en.wikipedia.org/wiki/List_of_regular_polytopes#Hyperbolic_tilings)
Perhaps this is the way to approach it, by looking at tesselations in this way, and starting at the simplest case and working out how to expand it into more complex ones. 
Anyway way, that's just my initial thoughts. :)
