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 Post subject: 4x4x4x4x4x4 Sudoku CubePosted: Mon Dec 22, 2008 3:54 pm

Joined: Wed Nov 19, 2008 3:02 am
The more complete title of this post should be:
(4x4x4)cubies x 4 base colors x 4 sticker colors x 4 symbols

The 4x4x4 x 4 Colors- Cube

A while ago I peeled all the original stickers off a Rubik’s Revenge and replaced them with stickers of only four different colours. (see Fig. 1 and Fig. 2).
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Fig1.jpg [ 48.28 KiB | Viewed 4043 times ]
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Fig2.jpg [ 51.18 KiB | Viewed 4041 times ]

This cube can now be brought to a position where every face is covered by a different complete and correct 2x2 Sudoku. Actually a “supersudoku” where on both diagonals all four colors appear exactly once. (Fig.3)
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Fig3.jpg [ 50.93 KiB | Viewed 4035 times ]
It is also possible to have an arrangement where the other six of the twelve possible 2x2 supersudoku solutions appear.
There is an fast and easy way to transform Fig.1 into Fig. 3. For now I don’t want to spoil the fun of finding it by giving more details. I just call this transformation “T” so I can refer to it further down.

An open question: How many cubes would you need to display all 78 possible (and distinguishable) 2x2 Sudoku solutions simultaneous? Between 13 and 68 cubes is the closest to the answer I got so far.
Then I read the post by Volitar Prime
“ 4x4x4 Sudoku Cube using colors instead of numbers „
http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=11306&start=0&st=0&sk=t&sd=a&hilit=4x4x4+sudoku

Inspired by this I thought again about sudoku on a 4x4x4 cube.

The 4x4x4 x 4 Colors x 4 Symbols- Cube

Instead of using 16 colors, I used 4 colors and added 4 symbols (numers 1 to 4) resulting also in 16 different possibilities for a facelet. (Fig. 4)
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Fig4.jpg [ 57.54 KiB | Viewed 4034 times ]

But how do these have to be arranged to match the Sudoku rules as stated by Volitar?

„...In order to apply the sudoku rules across all surfaces of the puzzle I used a 4x4x4 cube and 16 numbers instead of 9. This allowed me to have all 16 numbers on each of the 6 faces, without duplicates. Also when you follow any column or row all the way around the puzzle, you will also find all 16 numbers....“

If you place the 4 numbers (symbols) in such a way on the cube that in every corner (2x2x2) of the cube in Fig. 1 the numbers in themselves comply on every corner individually with the rules given above, than you have automatically a complete cube that complies with the 4x4 Sudoku rules as stated before. The sudoku solution for every corner can be the same or every corner can be solved different. That means this is a way for finding millions of different valid solutions for placing the stickers on the cube. Fig. 5 shows an example.
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Fig5.jpg [ 57.3 KiB | Viewed 4027 times ]

The numbers in Fig. 5 are placed so that the numbers fulfill two additional conditions 1) all the six faces show a solved 2x2 sudoku
2) when “T” (mentioned above) is applied to the cube, the result (Fig. 6) is a state where the base-colors on all faces form a 2x2 Sudoku and the color-symbol combination a 4x4 sudoku on the whole cube.
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Fig6.jpg [ 53.61 KiB | Viewed 4019 times ]

Finally the 4x4x4 x 4 Colors x 4 sticker colors x 4 Symbols- Cube

Starting with a (4x4x4) x 4 Colors- Cube in supersudoku configuration, I noticed that one can apply a second set of stickers to the center of the facelets (again four different colours), so that the following statements are true:
1) on all 6 cubefaces all 16 basecolor-stickercolor combinations are present
2) on every cubie there is only one colour of secondary stickers
3) the secondary stickers show 2x2 supersudokus on all 6 faces
(Fig.7)
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Fig7.jpg [ 54.21 KiB | Viewed 4015 times ]

Putting it all together one ends up with a cube where every facelet carries markings on three levels: basecolor-sticker color- symbol
The cube can be arranged so any of the three levels is grouped ( identical colors/symbols close to each other), it is also possible to bring it to states where any of the three levels show 2x2 sudokus on all 6 faces.
Finally the basecolor – symbol combination allows for 4x4 sudokus on the whole of the cube.
(Euler squares can also be found all over the place.)
I do not have the stickers which I think would look good and have all of the above. I put them together though on a GIF that can be used in “Cube Twister” (Fig.8, Fig.9, Fig.10)
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Fig8.gif [ 85.63 KiB | Viewed 1975 times ]
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Fig9.jpg [ 50.4 KiB | Viewed 4008 times ]
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Fig10.jpg [ 51 KiB | Viewed 4013 times ]

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 Post subject: Re: 4x4x4x4x4x4 Sudoku CubePosted: Mon Dec 22, 2008 8:51 pm

Joined: Fri Dec 12, 2008 12:16 am
Wow! Love how the stickers evolved.

_________________
PB's:
2x2: 6.17 (lucky)| 3x3: 29.79 (Extremely Lucky)
4x4: 3:18.62 | 5x5: 8:22.90
Megaminx: 9:01:94| 7x7: 39:51.98
Magic: 1.14 | Pyraminx: 9.76
2x2-5x5 Relay: 16:43.51

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 Post subject: Re: 4x4x4x4x4x4 Sudoku CubePosted: Fri Jan 16, 2009 2:59 pm

Joined: Sat Jan 28, 2006 6:04 am
Location: Switzerland
In collaboration with Ortwin I created a 4x4x4x4x4x4 Sudoku Cube page where you can find more informations.

This amazing cube can be solved in several '2x2 Sudoku' stages with several valid solutions:

Solved State A (Initial State)
1) Base colors are grouped into four 2x2 boxes on all six faces.
2) Circle colors are arranged in '2x2 Sudoku' style on two faces.
3) Symbols are arranged in '2x2 Sudoku' style on all six faces.
Note: The base color-symbol combination is representing a '4x4 Sudoku'.

Solved State B
1) Base colors are arranged in '2x2 Supersudoku' style on all six faces.
2) Circle colors are arranged in '2x2 Supersudoku' style on all six faces.
3) Symbols are arranged in '2x2 Supersudoku' style on two faces. Symbols are grouped into four 2x2 boxes on two faces.
Note: The base color-symbol combination is representing a '4x4 Sudoku'.

Solved State C1
1) Base colors are arranged in '2x2 Sudoku' style on two faces.
2) Circle colors are grouped into four 2x2 boxes on all six faces.
3) Symbols are arranged in '2x2 Sudoku' style on two faces. Symbols are grouped into four 2x2 boxes on two faces.
Note: The base color-symbol combination is representing a '4x4 Sudoku'.

Solved State C2
1) Base colors are arranged in 2x2 Supersudoku style on all six faces.
2) Circle colors are grouped into four 2x2 boxes on all six faces.
3) Symbols are arranged in 2x2 Supersudoku style on two faces. Symbols are grouped into four 2x2 boxes on two faces.
Note: The base color-symbol combination is representing a '4x4 Sudoku'.

Solved State D
1) No remarkable arrangement of the base colors.
2) No remarkable arrangement of the circle colors.
3) Symbols are gouped on all six faces.

Note
Positions where base colors, circle colors and symbols are brought all together either into '2x2 Sudoku' or '2x2 Supersudoku' positions are not possible.

The following terms are helpful to understand the concept of this cube:

2x2 Sudoku
When on a cube face all four elements of the same group show exactly once on every row and column.

2x2 Supersudoku
When on a cube face all four elements of the same group show exactly once on every row and column as well as on both diagonals.

4x4 Sudoku
When on all six cube faces the combination of two group elements show exactly once. In addition, each of the twelve rows (4 rows per axis), consisting of 16 stickers (along four faces of the cube), must show each combination exactly once as well.

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 Post subject: Re: 4x4x4x4x4x4 Sudoku CubePosted: Sun Jan 25, 2009 11:51 am

Joined: Mon Oct 13, 2008 12:48 am
Hi Ortwin,

That's a fantastic design.

I have added now your 4x4x4x4x4x4 Sudoku Cube to the default template of CubeTwister.

For this purpose, I have renamed it to 4^6 Sudoku Cube because this looks easier on the eyes than 4x4x4x4x4x4 but gives the same result - at least in mathematical terms.

I hope you don't mind. I can change the title back in case you don't like it. I left the full title in the notes about the cube.

Cheers,

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