# Is this 3d Sudoku possible?

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).

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.

• @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!)
– JMP
Jun 18, 2019 at 16:04
• 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? Jun 18, 2019 at 16:22
• Is the condition just that each (orthogonal) plane is a regular 2d Sudoku or is there anything 3d going on in the whole puzzle? Jun 18, 2019 at 16:23
• @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.
– JMP
Jun 18, 2019 at 16:28
• 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.
– Moti
Jun 19, 2019 at 15:45

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


• Wow. Very long solution. Apr 15, 2020 at 15:13