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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: this, (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.

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  • $\begingroup$ @hexomino; I am trying to construct a 9*9*9 number grid which is a Sudoku along all planes, but first I thought I'd ask for opinions on whether this is possible or not (the number of givens bit is a bonus!) $\endgroup$ – JonMark Perry Jun 18 at 16:04
  • $\begingroup$ Could you clarify what a 9x9x9 Sudoku is actually meant to be? In an ordinary 2d Sudoku you have rows, columns and cells, each of size 9. In a 9x9x9 you have rows and columns of size 9, faces of size 81, 3x3x3 cells of size 27 ... exactly what sets of cells do you want to have to contain one of everything? $\endgroup$ – Gareth McCaughan Jun 18 at 16:22
  • $\begingroup$ Is the condition just that each (orthogonal) plane is a regular 2d Sudoku or is there anything 3d going on in the whole puzzle? $\endgroup$ – Gareth McCaughan Jun 18 at 16:23
  • $\begingroup$ @GarethMcCaughan; if you take any planar slice of 9*9*1, this is the same form as a regular Sudoku, and needs to demonstrate this. Anything extra would be a bonus, but is not part of my question. $\endgroup$ – JonMark Perry Jun 18 at 16:28
  • $\begingroup$ The drawing kind of confusing - I understand that you mean that the 9X9X9 is constructed from 27 regular Sudoku's - right? Your image is not fully clear how to locate the numbers in the 3D 9X9X9. $\endgroup$ – Moti Jun 19 at 15:45
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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.enter image description here

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

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