|
These are Rubik's cubes of the form
3d, with the original popular puzzle being 33. We label the puzzles like this because they are a
d-dimensional cube broken into
3d smaller pieces or "cubies" of the same dimension. For example, the 3D cube has
33 or 27 total 3-dimensional cubies.
Each of the d-dimensional cubies could be considered to have its faces covered by stickers of one smaller (d-1) 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 1-colored, 2-colored, etc. pieces.
In general, a d-dimensional 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 d-dimensional 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:
2s . dCs
where,
d is the number of
dimensions,
s is the number of stickers on a given type, and
dCs 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 (2-colored pieces) on a
33 cube, we have s=2 and d=3 so,
number of edges =
22 . 3C2 = 4.( 3!/(2!(3-2)!) ) = 4.3 = 12
Here is a chart for the number of all the different piece types for
3d 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 3d
|
|
|
|
|
|
|
|
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 2-dimensional, 1-dimension, and 0-dimensional 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 nd and everything gets multiplied by a factor of
(n-2)(d-s). The full formula is:
2s .
dCs . (n-2)(d-s)
Note that this means the number of d-colored pieces
(corners) stay the same for any n since s=d in that case. Below are charts for
n4 and n5. If we had a 3D spreadsheet program, we could list out all the piece numbers for the more general
nd :)
Number of pieces of type s for
Rubik's type puzzle of the form n4
|
|
|
|
|
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 n5
|
|
|
|
|
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
|
|
