Sudokube

I get irrationally annoyed by things. Take a look at this festive stocking filler:

While this would be a diverting little puzzle, it isn't properly a Sudoku. Getting each face to have 1-9 may be more challenging than a traditional Rubik's Cube, it doesn't have that beautiful arrangement that Sudoku has. Indeed, it's rather ugly as each row is 12 squares whereas there are only 9 digits to play with.

To that end, I wondered if it would be possible to create a 4*4 faced puzzle cube. Such a cube could have the digits 1 - 16 on each face, row and column. What larks! I first thought about this two years ago, but never got very far with it.

If you can't visualise it in 3D space, here is how it would look flattened out. Lines have been added to show where the continuation points.

So, this looks fairly do-able, right? Wrong! While it's easy to start, it rapidly becomes more and more difficult to generate a solution to this cube. This is, if you will, a multidimensional Sudoku puzzle. But without the starting numbers.

I've had a look through various maths, puzzle and sudoku forums and I've yet to find any attempts at this - let alone a solution.

So is it possible and - if it is - will it be next year's big puzzle sensation?

Any help gratefully appreciated.

Update

Turns out there's a whole bunch of deep mathematics required to create such a puzzle. And, like everything on the Internet, one of the solutions has already been found.