These are Rubik's cubes of the form
3^{d}, with the original popular puzzle being 3^{3}. We label the puzzles like this because they are a
ddimensional cube broken into
3^{d} smaller pieces or "cubies" of the same dimension. For example, the 3D cube has
3^{3} or 27 total 3dimensional cubies.
Each of the ddimensional cubies could be considered to have its faces covered by stickers of one smaller (d1) dimension.
But each cubie also only exposes a subset of its stickers to the "outside", meaning these are the stickers you could see if you lived and operated in d dimensions.
We can use the number of exposed stickers as a classification of cubie types.
For the 3D case, the 27 cubies are broken into 4 types, those that expose 0 stickers, 1 sticker ("centers"), 2 stickers ("edges"), or 3 stickers ("corners").
Each sticker on a given cubie has its own color, so we could also call these 1colored, 2colored, etc. pieces.
In general, a ddimensional cube will have
d+1 of these types, those that expose 0,1,...,d different colored stickers.
By starting with the number of pieces in the 3D, 4D, and 5D puzzle versions, one could perhaps extrapolate a formula for the number of each of these types of pieces for any ddimensional Rubik's Cube (It was easier for me to arrive at this by looking at vector representations of
sticker coordinates on the 5D cube).
In any case, the formula for the number of a given type is:
2^{s .} _{d}C_{s}
where,
d is the number of
dimensions,
s is the number of stickers on a given type, and
_{d}C_{s} means the number of combinations of d things taken s at a time, equal to
d!
s!(d − s)!
For example, to get the number of edges (2colored pieces) on a
3^{3} cube, we have s=2 and d=3 so,
number of edges =
2^{2 .} _{3}C_{2} = 4^{.}( 3!/(2!(32)!) ) = 4^{.}3 = 12
Here is a chart for the number of all the different piece types for
3^{d} cubes up to dimension 10. There are all sorts of interesting patterns in here.
For example notice that the 5D cube has 2 types with the same number of pieces,
but this is not true in general for any dimension.
Number of pieces of type s for
Rubik's type puzzle of the form 3^{d}







s






total

total



0

1

2

3

4

5

6

7

8

9

10

pieces

stickers


0

1











1

0


1

1

2










3

2


2

1

4

4









9

12


3

1

6

12

8








27

54


4

1

8

24

32

16







81

216

d

5

1

10

40

80

80

32






243

810


6

1

12

60

160

240

192

64





729

2916


7

1

14

84

280

560

672

448

128




2187

10206


8

1

16

112

448

1120

1792

1792

1024

256



6561

34992


9

1

18

144

672

2016

4032

5376

4608

2304

512


19683

118098


10

1

20

180

960

3360

8064

13440

15360

11520

5120

1024

59049

393660

Another cool observation is that with this formula, we can go backwards and easily find the proper analogies for 2dimensional, 1dimension, and 0dimensional Rubik's cubes.
These aren't very exciting puzzles because none of them can actually be scrambled, but thinking about their cubie types is a little interesting.
Using the above, you can then also find
a piece counting formula for "Rubik's Revenge" (4 divisions per side instead of 3), "Professor Cube" (5 divisions per side), etc. versions of any d dimensional Rubik's Cube.
If n is the number of divisions per side, the puzzle has the form n^{d} and everything gets multiplied by a factor of
(n2)^{(ds)}. The full formula is:
2^{s .}
_{d}C_{s }^{. }(n2)^{(ds)}
Note that this means the number of dcolored pieces
(corners) stay the same for any n since s=d in that case. Below are charts for
n^{4} and n^{5}. If we had a 3D spreadsheet program, we could list out all the piece numbers for the more general
n^{d} :)
Number of pieces of type s for
Rubik's type puzzle of the form n^{4}




s



total

total



0

1

2

3

4

pieces

stickers


0









1









2

0

0

0

0

16

16

64


3

1

8

24

32

16

81

216


4

16

64

96

64

16

256

512

n

5

81

216

216

96

16

625

1000


6

256

512

384

128

16

1296

1728


7

625

1000

600

160

16

2401

2744


8

1296

1728

864

192

16

4096

4096


9

2401

2744

1176

224

16

6561

5832


10

4096

4096

1536

256

16

10000

8000

Number of pieces of type s for
Rubik's type puzzle of the form n^{5}




s




total

total



0

1

2

3

4

5

pieces

stickers


0










1










2

0

0

0

0

0

32

32

160


3

1

10

40

80

80

32

243

810


4

32

160

320

320

160

32

1024

2560

n

5

243

810

1080

720

240

32

3125

6250


6

1024

2560

2560

1280

320

32

7776

12960


7

3125

6250

5000

2000

400

32

16807

24010


8

7776

12960

8640

2880

480

32

32768

40960


9

16807

24010

13720

3920

560

32

59049

65610


10

32768

40960

20480

5120

640

32

100000

100000

