Revision as of 03:54, 20 June 2026

Number of permutations of 3x3

To count number of permutations of 3x3, count permutations due to both the placement and orientation of each type of piece (corners and edges; centers are fixed):

3 possible rotations of 8 corners, 7 corners determine the 8th = $$ 3^7 $$
2 possible rotations of 12 edges, but only 11 edges can be flipped independently; the orientation of the 12th edge is forced by the other 11 = $2^{11}$
8 different corner pieces to be distributed to 8 locations = $$ 8! $$
12 different edge pieces to be distributed to 12 locations = $$ 12! $$
Half of the cube permutations (those requiring an odd permutation of edges/corners) cannot be reached without disassembling the cube = $\frac{1}{2}$

Correction (April 2025): The edge-orientation factor was previously listed as $2^{12}$ . Only 11 of the 12 edges can be flipped independently; the final edge's orientation is determined by the parity of the other 11. This constraint is separate from the $\tfrac{1}{2}$ factor already applied for the even-permutation requirement. Fixing this brings the count from ~86 quintillion down to the correct value of ~43 quintillion, matching the widely published number (e.g., Wikipedia). The 5×5 section below already correctly notes this distinction.

Total number of permutations of a 3x3 cube:

$N = 3^{7} \times 8! \times 2^{11} \times 12! \times \frac{1}{2} = 43252003274489856000 \sim 4.32520 \times 10^{19}$

or about 43 quintillion permutations.

Number of permutations of 4×4

On a 4×4 cube (Rubik's Revenge), there are 8 corners, 24 edges, and 24 centers. Unlike the 3×3, the 4×4 has no fixed centers; all 24 orientations of the solved cube are equivalent.

Correction (June 2026): Three errors in the original 4×4 count have been fixed:

Edge wing placement: The original count used $\left(\frac{24!}{12!}\right)^2$ , treating the 12 left-wing and 12 right-wing edge pieces as each independently occupying 24 positions. However, after the 12 left wings are placed, only 12 slots remain for the right wings. All 24 edge pieces are distinguishable, so the correct factor is simply $$ 24! $$ (all 24 edge pieces arranged in 24 positions). This overcounted by a factor of about 2.7 million.
Center indistinguishability: The original count used $$ 24! $$ for centers, but the four center pieces of each color are visually identical. The correct factor is $$ 24! / (4!)^6 $$ (dividing by $$ 24^6 $$ , as there are $$ 4! = 24 $$ ways to arrange the four pieces of each of the 6 colors). This overcounted by a factor of $24^6 \approx 1.9 \times 10^8$ .
Missing orientation reduction: Because the 4×4 has no fixed centers, all 24 orientations of the solved cube are equivalent. The original count did not divide by 24.

These three corrections reduce the total from $\sim 9.18 \times 10^{61}$ to the widely published value of $\sim 7.40 \times 10^{45}$ — a difference of roughly 16 orders of magnitude.

Counting each factor:

Corners:

Placement: 8 corners being placed, for a total of $$ 8! $$
Orientation: 3 orientations are possible on the 8 corners, placement of 7 determines the last, for total of $$ 3^7 $$

Edges:

The 24 edge pieces are all distinguishable (each has a unique color pair, and corresponding edges are mirror images of each other). Any permutation is possible: $$ 24! $$
Edge pieces cannot be flipped due to their internal shape; orientation is determined by the position occupied.

Centers:

There are 24 center pieces. The four centers of each color are indistinguishable: $$ 24! / (4!)^6 $$

Orientation factor:

Because there are no fixed centers, all 24 orientations of the solved cube are equivalent. Divide by 24.

The total is:

$T = \frac{8! \times 3^7 \times 24!^2}{24^7} \approx 7.40 \times 10^{45}$

The full number is 7 401 196 841 564 901 869 874 093 974 498 574 336 000 000 000 (about 7.4 quattuordecillion).

This matches the published figure (e.g., Wikipedia's Rubik's Revenge article, and Singmaster's Cubic Circular Issues 3 & 4, 1982).

Counting Cycles

Length of Cycles on 3x3

One of the interesting features of a Rubik's Cube is that, starting from a solved cube, if any sequence of moves is repeated long enough on the cube, will eventually result in the cube returning to the solved state.

The simplest example is a single move (like U): after executing U four times, the cube is returned to the solved state.

It is a little less trivial with a sequence of moves like U'R'UR or LU, but these also eventually result in returning the cube to its solved state after 6 repetitions of the sequence U'R'UR or 105 repetitions of the LU sequence.

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 or about 43 quintillion permutations.
+==Number of permutations of 4×4==
+On a 4×4 cube (Rubik's Revenge), there are 8 corners, 24 edges, and 24 centers. Unlike the 3×3, the 4×4 has no fixed centers; all 24 orientations of the solved cube are equivalent.
+'''Correction (June 2026):''' Three errors in the original 4×4 count have been fixed:
+# '''Edge wing placement:''' The original count used <math>\left(\frac{24!}{12!}\right)^2</math>, treating the 12 left-wing and 12 right-wing edge pieces as each independently occupying 24 positions. However, after the 12 left wings are placed, only 12 slots remain for the right wings. All 24 edge pieces are distinguishable, so the correct factor is simply <math>24!</math> (all 24 edge pieces arranged in 24 positions). This overcounted by a factor of about 2.7 million.
+# '''Center indistinguishability:''' The original count used <math>24!</math> for centers, but the four center pieces of each color are visually identical. The correct factor is <math>24! / (4!)^6</math> (dividing by <math>24^6</math>, as there are <math>4! = 24</math> ways to arrange the four pieces of each of the 6 colors). This overcounted by a factor of <math>24^6 \approx 1.9 \times 10^8</math>.
+# '''Missing orientation reduction:''' Because the 4×4 has no fixed centers, all 24 orientations of the solved cube are equivalent. The original count did not divide by 24.
+These three corrections reduce the total from <math>\sim 9.18 \times 10^{61}</math> to the widely published value of <math>\sim 7.40 \times 10^{45}</math> — a difference of roughly 16 orders of magnitude.
+Counting each factor:
+'''Corners:'''
+* Placement: 8 corners being placed, for a total of <math>8!</math>
+* Orientation: 3 orientations are possible on the 8 corners, placement of 7 determines the last, for total of <math>3^7</math>
+'''Edges:'''
+* The 24 edge pieces are all distinguishable (each has a unique color pair, and corresponding edges are mirror images of each other). Any permutation is possible: <math>24!</math>
+* Edge pieces cannot be flipped due to their internal shape; orientation is determined by the position occupied.
+'''Centers:'''
+* There are 24 center pieces. The four centers of each color are indistinguishable: <math>24! / (4!)^6</math>
+'''Orientation factor:'''
+* Because there are no fixed centers, all 24 orientations of the solved cube are equivalent. Divide by 24.
+The total is:
+<math>
+T = \frac{8! \times 3^7 \times 24!^2}{24^7} \approx 7.40 \times 10^{45}
+</math>
+The full number is 7 401 196 841 564 901 869 874 093 974 498 574 336 000 000 000 (about 7.4 quattuordecillion).
+This matches the published figure (e.g., Wikipedia's ''Rubik's Revenge'' article, and Singmaster's ''Cubic Circular'' Issues 3 & 4, 1982).
 =Counting Cycles=

Rubiks Cube/Numbers: Difference between revisions

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