Is this 3d Sudoku possible?

Question

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.

Answer 1

The answer is

Yes, it is possible

Moreover,

You can use the solution to the $3 \times 3 \times 3$ puzzle to generate a solution.

First note that

If we take any set of three parallel $3 \times 3$ slices of the $3 \times 3 \times 3$ puzzle and permute them, we generate another valid $3 \times 3 \times 3$ grid. This observation will be the basis for our solution.

Step 1

Determine the solution to the $3 \times 3 \times 3$ grid.
Moti has already done this but here it is reproduced.

Step 2

Generate a $3 \times 3 \times 9$ grid by stacking this $3 \times 3 \times 3$ block on top of two more such block whose vertical-row planes are permutations of the original. If we read this grid from the top layer down (left to right), this looks as follows

 123     456     789     564     978     231     897     312     645

 564     978     231     897     312     645     123     456     789

 897     312     645     123     456     789     564     978     231

Notice that to get the 4th layer, for example, I've just rotated the rows in the 1st layer, same for 5th and 2nd, etc.

Step 3

Using this $3 \times 3 \times 9$ grid generate a $9 \times 3 \times 9$ grid with the new blocks being formed by permuting the vertical-column planes of the original blocks. If we read this grid from the top layer down (left to right), this looks as follows

 123     456     789     564     978     231     897     312     645

 564     978     231     897     312     645     123     456     789

 897     312     645     123     456     789     564     978     231

 ---     ---     ---     ---     ---     ---     ---     ---     ---

 312     645     978     456     897     123     789     231     564

 456     897     123     789     231     564     312     645     978

 789     231     564     312     645     978     456     897     123

 ---     ---     ---     ---     ---     ---     ---     ---     ---

 231     564     897     645     789     312     978     123     456

 645     789     312     978     123     456     231     564     897

 978     123     456     231     564     897     645     789     312

Step 4

Using this $9 \times 3 \times 9$ grid generate the $9 \times 9 \times 9$ Sudoku with the new blocks being formed by permuting the horizontal planes of the original blocks, in groups of three, and being placed adjacent. I will represent the full solution in three parts (as it is quite big):

Top three layers (first on the left, second in middle, third on the right)

 123|456|789          456|789|123          789|123|456

 564|978|231          978|231|564          231|564|978

 897|312|645          312|645|897          645|897|312

 ---+---+---          ---+---+---          ---+---+---

 312|645|978          645|978|312          978|312|645

 456|897|123          897|123|456          123|456|897

 789|231|564          231|564|789          564|789|231

 ---+---+---          ---+---+---          ---+---+---

 231|564|897          564|897|231          897|231|564

 645|789|312          789|312|645          312|645|789

 978|123|456          123|456|978          456|978|123

Middle three layers

 564|978|231          978|231|564          231|564|978

 897|312|645          312|645|897          645|897|312

 123|456|789          456|789|123          789|123|456

 ---+---+---          ---+---+---          ---+---+---

 456|897|123          897|123|456          123|456|897

 789|231|564          231|564|789          564|789|231

 312|645|978          645|978|312          978|312|645

 ---+---+---          ---+---+---          ---+---+---

 645|789|312          789|312|645          312|645|789

 978|123|456          123|456|978          456|978|123

 231|564|897          564|897|231          897|231|564

Bottom three layers

 897|312|645          312|645|897          645|897|312

 123|456|789          456|789|123          789|123|456

 564|978|231          978|231|564          231|564|978

 ---+---+---          ---+---+---          ---+---+---

 789|231|564          231|564|789          564|789|231

 312|645|978          645|978|312          978|312|645

 456|897|123          897|123|456          123|456|897

 ---+---+---          ---+---+---          ---+---+---

 978|123|456          123|456|978          456|978|123

 231|564|897          564|897|231          897|231|564

 645|789|312          789|312|645          312|645|789