For a more mathematical description, the game board is a graph, whose vertices are the individual squares and which has edges joining two vertices if they're opposite faces in some net. For the puzzle to be solvable, each connected component of this graph must be 2-colorable (which is easy to test), and in that case, the colorings (which are easy to generate) plus a starting number in each connected component easily give you the numbers for the entire board.
Again, this was probably the best puzzle game you could make about nets of dice, but I think the concept simply doesn't support a puzzle game. There's simply not enough interaction between the numbers.
It was an admirable attempt, but the concept simply doesn't translate into a puzzle game, or at least, not a non-trivial one.
Basically, the puzzle consists of filling in three unrelated sets of numbers: the 1's and 6's, the 2's and 5's, and the 3's and 4's. Each group only informs about and forces the locations of others in the group, so you can deal with each set completely separately. So you start with the 1's and 6's, say. You locate a 1 or a 6 on the field. You identify a net it's in, you deduce where the paired location in the net is (the one that would be opposite it if you folded it up into a cube), and you fill in the appropriate other number. Rinse and repeat until every net has a 1 and a 6. Do the same for the 2's and 5's, and then the same for the 3's and 4's. This will always solve the puzzle if it has a unique solution.
 "Dynetzzle"){kind=icon}