# Kilominx God's number bounds

#### xyzzy

##### Member
tl;dr: between 16 and 46 18 and 34 moves (inclusive).

The kilominx has (19!/2) ⋅ 3^18 ~ 24 septillion states (with a fixed corner), which is well beyond what we can exhaustively enumerate. Even with full symmetry+antisymmetry reduction, that's still around 98 sextillion states, which is a lot larger than the number of 3×3×3 states without any symmetry reduction. (You know a number is large when the spelling checker thinks that it's not a real word.)

This first post will be about an upper bound. (Lower bound analysis to come later, but for now you can have the "trivial" lower bound of 16 moves by counting move sequences without immediate cancellations.) We divide solving the kilominx into two phases, given by the following chain of subgroups: $$G_0:=\langle\text{all 12 moves}, \text{rotations}\rangle$$, $$G_1:=\langle U,R,F,D,DB,DBL\rangle$$, $$G_2:=\langle\rangle$$. For this, we do not use a fixed corner, so $$G_0$$ includes all 20!/2 even permutations of the corners. Further, while rotations can be done solely with face turns, keeping the "core" fixed, we explicitly include them in the generating set and treat them as free moves (i.e. rotations don't contribute to the move count).

Phase 1: separation

Note that $$G_1=\langle U,R,F,D,DB,DBL\rangle=\langle U,R,F\rangle\times\langle D,DB,DBL\rangle$$. This effectively divides the puzzle into two disjoint "hemispheres" that don't interact, centred at the U-R-F and D-DB-DBL corner pieces. For convenience, call these the north and south hemispheres, respectively. The right coset space $$G_1\backslash G_0$$ can be treated as a vector of 20 bits corresponding to whether each corner piece belongs to the south hemisphere in the solved state, along with an additional value in $$\mathbb Z/3$$ corresponding to the sum of the orientations of the 10 pieces belonging to the south hemisphere. In other words, this phase is equivalent to separating the pieces into the hemispheres they belong to, and ensuring that the corner orientation within each hemisphere adds up to 0.

EDIT: This is slightly wrong because I forgot about parity, but I don't have time to fix it right now. See qqwref's post later in this thread. The overall upper bound is unaffected, but the average move count increases slightly.

The coset space has $$\binom{20}{10}\cdot3=554268$$ cosets, but there is no need to consider the whole coset space. Firstly, we don't even need to care about corner orientation here; once the corners are separated into the hemispheres, we can change the total corner orientation within each hemisphere (without affecting the separation) by just rotating around the U-R-F corner. Secondly, it doesn't matter whether the pieces are all moved to the correct hemispheres; we could also have them all moved to the wrong hemispheres and then rotate the puzzle to put them in the correct hemispheres. This reduces the number of cases to consider to $$\binom{20}{10}/2=92378$$.

depth

# reduced cosets

0

10

1

240

2

4050

3

26890

4

52552

5

8616

6

20

Average: 3.707
Upper bound: 6

Phase 2: two 3-gen solves

The $$\langle U,R,F\rangle$$ subgroup, while much smaller than $$G_0$$, is still pretty large, having 36 billion states. It's small enough that a full breadth-first search can be done if symmetry+antisymmetry reduction is used, but I will leave this for another time.

Anyway, we can split this into solving orientation then permutation; for this, I used the same definition of orientation as in my solver. Unlike on a 2×2×2, it's not possible to define orientation in terms of a subgroup generated by single moves (e.g. you can't just say something like "the orientation defined by ⟨U, R2, F2⟩"). Here, we choose reference stickers on each of the corner pieces by taking the one that is closest to perpendicular to the U-D axis; equivalently, corner orientation is preserved by U, (R' F R F')3, flip and y. (There's a different choice of reference stickers that is more symmetrical when restricted to $$\langle U,R,F\rangle$$, which would make implementing symmetry/antisymmetry reduction easier.)

Orientation:

depth

# cosets

0

1

1

8

2

61

3

452

4

2674

5

9182

6

7070

7

235

Average: 5.19
Upper bound: 7

Permutation:

depth

# states

0

1

1

4

4

6

5

56

6

190

7

746

8

4727

9

25251

10

140966

11

602085

12

944543

13

95825

Average: 11.510
Upper bound: 13

(The distribution for the orientation substep is done with a standard breadth first search, but the distribution for the permutation step was computed by doing an IDA* search on each of the 10!/2 states with pieces already oriented. It's not possible to use a BFS to directly compute the latter distribution, as most moves disturb corner orientation.)

This gives an upper bound of 20 moves for $$\langle U,R,F\rangle$$, and since $$G_1$$ is just two copies of $$\langle U,R,F\rangle$$, the upper bound for phase 2 as a whole is 40 moves.

To conclude, this gives an upper bound of 46 moves for solving a kilominx.

depth

# states

0

0

1

0

2

0

3

1

4

0

5

3

6

10

7

87

8

671

9

5453

10

40232

11

245288

12

600187

13

108065

14

3

Average: 11.763

These are the three 14-move states (or, well, the inverses of these, but same thing):
U R U2' R2' F' U2 F' U2' R F2 R2 U' R2 U'
U R2 U2 R2 F' U R F' R2' F2' U2' R2' U' F2
U R U2' F2' U2' R F' R F U2 R2 F2' R2 U2'

These can probably be solved in fewer moves if we allow moves other than just U, R and F. (I don't have a full optimal solver yet.)
Ignoring orientation, there are 284 permutations at depth 9 (and none at depth 10). Solving all 284×3^9 states with those permutations, we get this distribution:

depth

# states

9

18648

10

175957

11

1239372

12

3447704

13

707850

14

441

Average: 11.832

The proportion of 14-movers in this specially chosen sample is much higher than in a uniform random sample. All of these 14-movers are local maxima, and most are "strict" local maxima (in the sense that every legal move brings it to depth 13). Up to symmetry, there's one 14-mover among these that isn't a strict local maximum, and it's one move away from a 14-mover not among those; this new 14-mover is also a local maximum and isn't adjacent to any other 14-movers.

This very strongly suggests that the actual diameter should be 14 moves, which would bring the overall upper bound down to 6+14+14=34 moves.

On another note, my current kilominx random state scrambler uses different solving phases and has a move count of 27.8±1.3 moves. The scrambling/solving is done with "hemisphere notation", which includes the U, R, F, L, BL and BR moves, in addition to flipping the whole puzzle over (i.e. rotating 180 degrees around the left-right axis), denoted as either "x2" or "flip". Since scramble sequences are meant to be applied by human scramblers, minimising rotations was a design goal, and the solving method used outputs a maximum of two flips per scramble. (Furthermore, we want to avoid "weird" rotations, because that would require additional scrambling notation, and the simpler the notation, the less likely scrambling mistakes are; to that end, flipping is the only type of rotation used.)

While the phases used in this post would lead to a slightly lower average move count (3.707+11.763+11.763 = 27.233 moves), the first phase makes use of arbitrary rotations while separating the pieces, which is extremely undesirable.

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#### jaap

##### Member
Did you count (R' F R F')3 as 4 moves?

#### xyzzy

##### Member
Did you count (R' F R F')3 as 4 moves?
That move sequence would be counted as 12 moves. (Not actually used in the solver, but to demonstrate the moves that preserve corner orientation.)

#### jaap

##### Member
Ok. I was a bit confused by your permutations calculation, as it has to preserve orientations, but presumably then you didn't just use a breadth first search using the orientation preserving generators you listed. Did you do a breadth first search using single moves and just filter from the output the ones that had the wrong orientation? Or did that take too long and you had to tackle the deeper positions separately with separate searches?

#### xyzzy

##### Member
Ok. I was a bit confused by your permutations calculation, as it has to preserve orientations, but presumably then you didn't just use a breadth first search using the orientation preserving generators you listed. Did you do a breadth first search using single moves and just filter from the output the ones that had the wrong orientation? Or did that take too long and you had to tackle the deeper positions separately with separate searches?
Right, I should've elaborated on this. I solved all of the 10!/2 positions with oriented pieces with IDA*. For example, this is one of the solutions produced: F' R' F2' R2' U2 F2' U2 R U2 R2' U. Each solve takes ~4 milliseconds, so this "brute force" approach took only around two hours to complete.

#### jaap

##### Member
Cool. Anyway, this is really good work. I like the clever reductions due to the rotations of the whole puzzle.

Have you tested the neighbours (and neighbours of the inverses) of the three depth 14 positions you found in <U,R,F>? It's unlikely to result in a depth 15 position, but it is worth checking.

#### xyzzy

##### Member
Have you tested the neighbours (and neighbours of the inverses) of the three depth 14 positions you found in <U,R,F>? It's unlikely to result in a depth 15 position, but it is worth checking.
Oh, I didn't think of that. Just ran the solver on the neighbours, and those three positions and their inverses are local maxima (every move brings them to depth 13).

I like the clever reductions due to the rotations of the whole puzzle.
Funny thing is, I originally had this idea for megaminx, where you cannot use these "automatic symmetry reductions" because of the fixed centres. (And on a megaminx, G_1\G_0 is a much larger coset space because of the edges.) Still trying to figure out what's a good way of upper-bounding megaminx god's number that isn't just "build blocks, then more blocks, etc. etc.". I have some vague ideas, but they would be tricky to implement.

#### qqwref

##### Member
Ben Whitmore recently computed God's Algorithm on the <U,R,F> subgroup and God's Number is 14. So the whole puzzle can be solved in at most 34 moves.
http://cubezzz.dyndns.org/drupal/?q=node/view/560

EDIT: Hey xyzzy, if you see this, it doesn't sound like you considered permutation parity in the separation step. That is, each half must have even permutation. Can you try doing the computation taking that into account? One way might be, for each coset, continue searching for separation algorithms until you find two where the permutation parities of the "top hemisphere" pieces differ. Then one of those two will solve one permutation-parity version of that coset and the other will solve the other, even though you don't know which is which. You can check for differing hemisphere permutations on a puzzle with all pieces distinct, but the search can still be done in the coset space.

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• DGCubes

#### xyzzy

##### Member
it doesn't sound like you considered permutation parity in the separation step.
Oh dang, you're right. There's a way to incorporate the permutation parity into the separation directly and I've done this before (for the Redi Cube solver/scrambler). I don't know why I suddenly just forgot about it…

Will rerun the code and post results shortly.

Edit: Take 2! This is the fixed move distribution for the separation step, now taking parity into account:

depth

# reduced cosets

0

20

1

480

2

9480

3

86960

4

223808

5

48684

6

80

Average: 3.841
Upper bound: 6

Upper bound remains unchanged, which is nice to know. I didn't feel like wrapping my head around whether some of the symmetry I exploited in the original post was valid with parity taken into consideration, so this only does the "corner twist" symmetry reduction and has $$\binom{20}{10}\cdot2$$ reduced cosets.

Also interesting to note is that the depth-6 positions are the same regardless of whether parity is taken into account. (The original (incorrect) table lists 20 reduced cosets and this one lists 80 reduced cosets; this factor-of-four difference comes from using less symmetry reduction (2) and including parity (2).)

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• muchacho, qqwref and DGCubes

#### qqwref

##### Member
Thanks, cool stuff! Ben Whitmore has also computed <R,U,F> in qtm:

Depth

New

Total

0

1

1

1

6

7

2

30

37

3

144

181

4

696

877

5

3360

4237

6

16176

20413

7

77520

97933

8

369444

467377

9

1752204

2219581

10

8254390

10473971

11

38490087

48964058

12

176067961

225032019

13

772274582

997306601

14

3062170056

4059476657

15

9439992329

13499468986

16

15628156836

29127625822

17

6433416584

35561042406

18

151790551

35712832957

19

2243

35712835200

Would you mind computing separation in qtm as well?

#### xyzzy

##### Member
Clearly we should unban Ben and let him post here directly…

My BFS code doesn't directly support "qtm" but it shouldn't be too hard to add that. (In class now; will do later.)

Edit: Separation + parity, qtm.

depth

# reduced cosets

0

20

1

240

2

2460

3

16640

4

69644

5

144740

6

119688

7

160000

8

80

Combined with Ben's upper bound of 19q for ⟨R,U,F⟩, this gives a bound of 46q in total. (Incidentally, the same as my original bound for FTM…) It'd be nice to get independent verification for this, since I haven't tested this BFS function in an actual solver.

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• DGCubes

#### xyzzy

##### Member
And now, a slightly better lower bound.

Assign this ordering to the twelve face moves of a kilominx: U, DFL, DBL, BL, DB, F, L, DFR, R, BR, DBR, D. (So U < DFL < DBL < … < DBR < D.) We keep track of the last move applied and count the number of move sequences of a given length with a given last move (if the length is at least 1), and reject all move sequences where moves cancel or we can reduce its lexicographic order (wrt the above ordering on moves) by swapping two consecutive moves that commute.

This excludes, for example, "U D U" (the second and third moves can be swapped, then the first and second moves cancel) and "U R D" (the "R" and "D" can be swapped). This almost provides a canonicalisation procedure for move sequences, but it's possible that a move sequence can have multiple reductions, none of which can be reduced further. (This doesn't happen on n×n×n puzzles with OBTM because whether moves commute forms an equivalence relation there; otoh, on the dodecahedral puzzles, you can have three moves such that the first commutes with the second, the second commutes with the third, but the first and third don't commute. An example is L, R, BL.)

If you work out the whole thing, it's a pretty straightforward linear recurrence in 12 variables (one variable for each possible "last move"), although some care is needed in handling the distance-0 and distance-1 special cases.

depth

# reduced move sequences

0

1

1

48

2

1536

3

44928

4

1293568

5

37138432

6

1065635840

7

30573592576

8

877152370688

9

25165283983360

10

721985189445632

11

20713556630568960

12

594266228722237440

13

17049334158528086016

14

489140692981253144576

15

1.4033311521445843e+22

16

4.026118355634329e+23

17

1.155082247599007e+25

18

3.3138990979964524e+26

19

9.507485076950928e+27

Modulo rotations, a kilominx has 19!×3^18/2 ~ 2.36 × 10^25 states, and since there are fewer move sequences up to length 17 than that, it follows that God's number must be at least 18.

• DGCubes

#### Herbert Kociemba

##### Member
If you work out the whole thing, it's a pretty straightforward linear recurrence in 12 variables (one variable for each possible "last move"), although some care is needed in handling the distance-0 and distance-1 special cases.
I did an analysis for Megaminx https://www.speedsolving.com/forum/threads/lower-bound-for-megaminx-in-htm-and-qtm.35724/ some time ago. I see no reason why this argumentation should not hold for Kilominx since it has the same set of commutating and not-commutating moves. Especially the number of positions in depth 3 of Megaminx is exactly 43520 (Tom Rokicki provided this number) and this is also the number of "canonical" move sequences I got. So if you have 44928 move sequences for depth 3 with Kilominx it seems that your analysis is not complete. Also your asymptotic branching factor seems to be > 28 which is higher than the 26.4803 I got (which is in accordance with the number for the Dodecahedron given here ).

I get 3.7871*10^24 move sequences with 17 moves and 1.00283*10^26 with 18 moves, so unfortunately the lower bound still stays 18....

#### xyzzy

##### Member
I did an analysis for Megaminx https://www.speedsolving.com/forum/threads/lower-bound-for-megaminx-in-htm-and-qtm.35724/ some time ago. I see no reason why this argumentation should not hold for Kilominx since it has the same set of commutating and not-commutating moves. Especially the number of positions in depth 3 of Megaminx is exactly 43520 (Tom Rokicki provided this number) and this is also the number of "canonical" move sequences I got. So if you have 44928 move sequences for depth 3 with Kilominx it seems that your analysis is not complete.
This is entirely correct and I don't know how I missed it!

I was working with an "almost-canonicalisation" procedure like this: for every pair (A, B) of commuting moves, decide whether AB or BA should be the "canonical" representation, and reject every move sequence with a "non-canonical" subsequence. The problem with this is that it's not truly canonical, which explains the discrepancy in our numbers. (The reason that this procedure cannot be canonical in the general case is that, if it were, the numbers would be independent of the ordering of moves I chose (U < DFL < …), but they're not independent.)

Also, something interesting I noticed was that, in the grand scheme of things, a branching factor of 26.something (your analysis) and a branching factor of 44 (naïve analysis) are almost the same on a logarithmic scale; in essence the more complicated analysis only manages to increase the lower bound from 16 moves to 18 moves.

#### Thom S.

##### Member
Clearly we should unban Ben and let him post here directly…
I don't think Ben would want his ben1996123 account back because the amount of lolben probably would still be big

#### Herbert Kociemba

##### Member
Also, something interesting I noticed was that, in the grand scheme of things, a branching factor of 26.something (your analysis) and a branching factor of 44 (naïve analysis) are almost the same on a logarithmic scale; in essence the more complicated analysis only manages to increase the lower bound from 16 moves to 18 moves.
This reminds me of a similar behavior concerning the more difficult to find God's number for all these puzzles which also seems to be only a few moves away from the easy to find lower bound, 20 moves versus 18 moves for Rubik's cube for example. This is intuitively quite clear. If you have for example a branching factor of 30 and the lower bound is N, then with N+1 moves you have more canonical sequences than the number of states and with N+3 or N+4 moves you have at least 900 times or 27000 times more canonical sequences than the number of states so that it not unlikely that you "hit" each state at least once.