The more complete title of this post should be:
(4x4x4)cubies x 4 base colors x 4 sticker colors x 4 symbolsThe 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
Fig1.jpg [ 48.28 KiB | Viewed 3963 times ]
Fig2.jpg [ 51.18 KiB | Viewed 3961 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)
Fig3.jpg [ 50.93 KiB | Viewed 3955 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
Fig4.jpg [ 57.54 KiB | Viewed 3954 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.
Fig5.jpg [ 57.3 KiB | Viewed 3947 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.
Fig6.jpg [ 53.61 KiB | Viewed 3939 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
Fig7.jpg [ 54.21 KiB | Viewed 3935 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
Fig8.gif [ 85.63 KiB | Viewed 1895 times ]
Fig9.jpg [ 50.4 KiB | Viewed 3928 times ]
Fig10.jpg [ 51 KiB | Viewed 3933 times ]