I was looking at images of 3d Sudoku's on Bing, because I was looking for a $9\times9\times9$ Sudoku, where each horizontal, vertical left-right and vertical top-bottom plane was also a Sudoku.
QUESTION: Is a $9\times9\times9$ Sudoku grid with every plane a 2d Sudoku possible?
The closest image relating to this question that I found was: , (supposedly from Tokfm but I couldn't find it there) which I solved (see Addendum).
Addendum
The question is NOT how to solve the $3\times3\times3$ image - this is fairy easy:
If two digits appear, then the third follows, because the first two occupy 2 coordinates in each of the x-y, x-z and y-z planes, leaving only one possible space (for example the two 5's are ({back,middle},{left, middle},{top, middle}), so the final 5 is (front, right, bottom)).
We can also see that the 6 on the top plane is in the middle, as the 6 already present operates along two of the plane orthogonal to the top plane.
but is asking for a proof/counter-example that a $9\times9\times9$ Sudoku grid with every plane a 2d Sudoku exists or not.
The comments contain more information as to what properties such a number grid would have.