Background

Readers of my blog may recall a tool I released in 2020 called bandaged cube explorer. It used javascript to visualize the state space of bandaged 3x3x3 puzzles.

I recently got an email from someone who had used it to look at 0x2000402849B5E. They pointed out that this graph has some vertices that are extremely far away from the starting configuration, and said “I wonder if there are bandaged puzzles with configurations that take even more moves to reach.”

I found this question fascinating and spent some time generating and analyzing data. I’d like to share some of my findings with you here.

Why this is interesting

In my 2020 blog post, I wrote “When I first tried to get into bandaged puzzles, I found a few docs online containing collections of named configurations […] What they both lacked was guidance on what kind of experience to expect when solving each puzzle”. To this day, I still don’t have a satisfying way to predict solving experience from bandaged signature.

Looking at the max distance between vertices has the potential to help with this. I suspect that puzzle where this metric is large contain long, highly constrained paths through the shape space, and will be a different solving experience from puzzles where this metric is small. To be clear, this is not about the difficulty of the solve, it’s about the experience of the solve, and the kinds of tools a solver will need to develop.

Also, I think no one has computed these numbers before, and that appeals to me as well.

Definitions

There are two common ways to measure how far apart two configurations are. In QTM (quarter turn metric), every quarter turn counts as one move. In HTM (half turn metric), a double turn of a single face only counts as one move. There are a few other interesting distance metrics but we will ignore those for this blog post.

In graph theory, we have a few terms to explain how far apart things are. A vertex’s eccentricity is the maximum distance from it to any other vertex. The vertex with the smallest eccentricity is called the center and its eccentricity is called the radius. Note that a graph may have more than one center. The largest eccentricity in a graph is called the diameter.

Techniques

Bandaged cube graphs are pretty tame by graph theory standards. All edges are bidirectional and have equal weight. This means we can do a breadth first search from each vertex in a graph and easily filter for the minimum and maximum eccentricities. As a bonus, we’ll get a representative example for each of those.

One useful optimization I found was to observe that bandaged configurations that only differ by orientation/mirroring must have the same eccentricity. Checking whether two cubes are the same modulo orientation is computationally cheap compared to breadth first search on most graphs so this improved runtime dramatically.

I applied this procedure to all 3559 bandaged configurations listed here and containing >1 vertex. Both the source code and the raw data are available on my github.

With all that out of the way, I’d like to share a few bandaged 3x3x3 configurations that are special in some way.

Cubes with very large diameter and radius

Largest

0x1003FE80000423 has by far the largest diameter, in both QTM and HTM.

Note that the ‘x’ over the white center means that this face is never allowed to turn. In QTM, 0x2003FD00041830 and 0x1003F680104444 are a whopping 167 moves apart (note that these two are mirror images of one another). This cube also has the largest diameter in HTM, 127.

I was quite surprised to see how small the graph for this configuration is. It has just 3780 vertices. Some configurations have over 20x as many vertices and yet have dramatically smaller diameters! (note: it’s still large enough to lag bandaged-cube-explorer. In this blog post, I’ll only deeplink to graphs that are small enough to render nicely on mobile).

At first, I found the bandaged white center on this cube a little unsatisfying, because my usual tools for playing with bandaged 3x3x3 puzzles (a cubetwist bandaged kit or the Magic Cube app) both can’t handle that. Then I checked the diameter of this configuration with the white center unlocked and found that it does not change!

Second largest

The configuration with the second largest diameter in both QTM and HTM is 0x83fe04000c21.

This configuration has a QTM diameter of 120 (example 0x83BE04021182 to 0x188c30c4204441) and an HTM diameter of 96.

This configuration also has a surprisingly small graph. It has a mere 2704 vertices.

Top 10 (QTM)

After this, QTM and HTM disagree about which configurations are largest. Here are the remainder of the top 10 in QTM:

1. 0x10000080800C63 (QTM: 117, HTM: 73)
2. 0x21109308C00C46 (QTM: 115, HTM: 90)
3. 0x108430C40007A3 (QTM: 113, HTM: 92)
4. 0x1003DA80000423 (QTM: 110, HTM: 90)
5. 0x1084308620063D (QTM: 107, HTM: 79)
6. 0x10039E80000403 (QTM: 101, HTM: 83)
7. 0x1084308420063D (QTM: 101, HTM: 75)
8. 0x21000102000C43 (QTM: 100, HTM: 59)

Top 10 (HTM)

And here is the rest of the top 10 in HTM:

1. 0x108430C40007A3 (#5 above)
2. 0x21109308C00C46 (#4 above)
3. 0x1003DA80000423 (#6 above)
4. 0x10039E80000403 (#8 above)
5. 0x1084308620063D (#7 above)
6. 0x72920080007F (QTM: 95, HTM: 77)
7. 0x1084308420063D (#9 above)
8. 0x139200800C1C (QTM: 94, HTM: 75)

The vertex counts of all these graphs are lower than I expected. The largest one is 0x1003DA80000423 with 7586 but most are well below 5000. To get a better sense of the data, I made this scatter plot of diameter (QTM) vs vertex count.

For more information on these configurations, such as examples of pairs of vertices that are the maximum distance apart, see the raw data.

Comparing QTM diameter to HTM diameter

Another thing I found surprising, was the different between QTM diameter and HTM diameter in configurations like 0x21000102000C43 (100 vs 59). Here’s a scatterplot of the raw data on these two measures:

High QTM/HTM

QTM diameter can never be larger than 2x the HTM diameter since each half turn can be replaced by two quarter turns. Over 50 configurations achieve the maximum of HTMx2 = QTM. The two that stand out to me are 0x8000802E3401 and 0x8000842FB421. Both have 110 vertices and a QTM diameter of 12. I suspect they’re actually the same puzzle in disguise as two different configurations.

Low QTM/HTM

QTM diameter can never be lower than HTM diameter. Almost 70 configurations have an equal QTM diameter and HTM diameter. This means that half turns are not useful when traveling between antipodes.

The largest of these by diameter (17) is a tie between 0x30138380002D and 0x80018620002D. The largest of these by vertex count are 0x180200021 (402, and a diameter of 15) and then 0x10000000000002, (240, and a diameter of 5).

A few other configurations that stood out to me are: 0x10000084018421 (2540 vertices, QTM diameter 18, HTM diameter 17) 0x108401843DCC21 (1130 vertices, QTM diameter 64, HTM diameter 56)

In a normal circle, the diameter is definitionally 2x the radius. In undirected graphs like ours, the diameter is capped at 2x the radius but can be smaller than that. Here’s a scatterplot of the overall relationship between the two:

There are ~70 puzzles where diameter and radius equal in QTM. By radius, the three largest ones are 0x30000010048883, 0x20108000000C03, and 0x30000000040203. All three have a QTM radius and diameter of 15. Amazingly, all three of these are familiar to me. The first one is the subject of a previous blog post of mine. The other two are both mentioned at the end of that blog post as potentially related configurations!

If we expand our search to puzzles where the diameter is only slightly larger than the radius then we can find some other puzzles whose graph forms one big loop.

One that jumped out at me was 0x80408B40DC7A1 (diameter 47 QTM, radius 38 QTM). This puzzle turns out to be the Maze-300 cube by André Kutepow (Isaev). It is the bandaged 3x3x3 with the largest known god’s number (301 QTM).

A few others that seem potentially interesting are 0x1108100440843 (diameter 48 QTM, radius 39 QTM) which has a beautiful spread out graph with lots of loops. Also 0x10843086218C21 (diameter 53 QTM, radius 42 QTM) and 0x139E0090002D (diameter 62 QTM, radius 46 QTM).

Is vertex count a good predictor of diameter?

Nope!

There are many examples of graphs with relatively low vertex count but large diameter, e.g. 0x21000100008C43 (726 vertices, QTM diameter of 95) and 0x100C00808005AD (482 vertices, 73 QTM diameter).

There are also plenty of examples of graphs with a high vertex count but relatively low diameter, such as 0x8000000804 (a.k.a “3 stripes”) (1296 vertices, QTM diameter of 8), and 0x20008000000806 (7344 vertices, QTM diameter of 14).

Looking at diameter / vertex count

I was hoping that this ratio might help me find some interesting puzzles but the results were disappointing. Configurations where this metric was high, such as 0x30080200000603 and 0x824810800C23 tended to have extremely constrainted graphs, (click through to see them). Their graphs have almost no branching and I suspect these puzzles can barely scramble.

Puzzles were this metric was small were ones with the largest vertex counts, and didn’t have particularly interesting diameters.

Closing thoughts

The maze 300 cube has a QTM diameter of only 48, many of the puzzles discussed in the blog post have a dramatically larger diameter. I wonder if one of these will be able to take the record for the configuration with the largest known god’s number.

If you find such a configuration, or you find another intersting feature in the data, please send me an email and let me know!

More thoughts - cube orientation

(This update was published on Jan 14, 2023.)

A user on the twisty puzzles forum raised an interesting point. The way my code computes graphs’ properties assumes that the orientation of the puzzle is fixed. This does not map nicely to how people visualize and solve puzzles. If two states differ only by oreintation, then a solver can use the same approach to both by just rotating the cube.

I re-ran my analysis but modified to consider two configurations the same if they only differ by orientation (or mirroring). I’ll start by looking at the results in isolation, and then do some comparison between these results and my previous results where cube orientation was considered.

When we ignore orientation, a few graphs shrink down to a single vertex and a diameter of 0. In case where dividing by 0 would be inconvenient, I’ll be ignoring these configurations instead.

If you’d like to play with the raw data, I made it available on my github. Putting all of it in one .csv felt unwieldly so I split this data into its own file. If you’d like to merge them, the rows are in the same order in both.

Largest vertex count

This has nothing to do with graph diameters but I’ve never seen these results posted before so I might as well mention them here. These are the four configurations with the largest vertex counts if we ignore orientation.

1. 0x10840080800423 (21490 vertices) [3rd most vertices when considering orientation]
2. 0x10041080200421 (18551 vertices) [7th most vertices when considering orientation]
3. 0x3DA00000C08 (16894 vertices) [10th most vertices when considering orientation]
4. 0x39E00000420 (15134 vertices) [13th most vertices when considering orientation]

Largest Diameter (ignoring cube orientation)

Unsurprisingly, many of the puzzles with a large diameter are familiar to us from earlier in this post. The puzzles with the largest diameter in QTM are:

1. 0x108430C40007A3 (113 QTM, 92 HTM) [5th largest QTM diameter with orientation considered]
2. 0x1003FE80000423 (103 QTM, 76 HTM) [1st largest QTM diameter with orientation considered]
3. 0x1003DA80000423 (99 QTM, 89 HTM) [6th largest QTM diameter with orientation considered]
4. 0x500180800C3D (91 QTM, 72 HTM)
5. 0x210B9100000863 (91 QTM, 71 HTM)
6. 0x10039A800005AD (86 QTM, 69 HTM)
7. 0x73920080007F (86 QTM, 68 HTM)
8. 0x1084308620063D (84 QTM, 64 HTM) [7th largest QTM diameter with orientation considered]
9. 0x18C400C7A00401 (84 QTM, 66 HTM)
10. 0x3DA00000C60 (83 QTM, 66 HTM)

Several puzzles on this list (#1, #4, #5, #6, #7, #9) have no self-symmetries, so their graphs are the same whether or not we consider orientation. The remaining puzzles all had 2x as many vertices before we stopped considering orientation.

0x10039A800005AD (#6) appeals to me for some reason. It’s graph is slightly too large to link but the shape is pretty fun; one very big loop and then a complex appendage coming off of it. It has quite a few “normal” edges which makes me think the solve will have some fun subproblems.

0x73920080007F (#7), has by far the smallest vertex count (518) of these puzzles. Its graph seems to contain no non-trivial loops. I suspect that when this puzzle is solved by shape, it is also solved by color.

Interestingly, the top 10 by HTM are the same but in a slightly different order.

QTM vs HTM diameter (ignoring cube orientation)

High QTM/HTM diameter

Again, several configurations achieve a the upper bound where QTM diameter is double HTM diameter. Most of these have a small diameter or seem otherwise boring. A few configurations get close and seem more interesting to me. Two that stand out are 0x1084008C238C63 (19 QTM, 10 HTM) and 0x10818086200C23 (39 QTM, 21 HTM). The former looks a lot like the cube with the largest vertex count but with a few bandaged centers.

Low QTM/HTM diameter

Many configurations achieve the lower bound where QTM and HTM diameters are equal. 0x80018620002D (discussed above) does not have any self symmetries so it is still the largest of these by diameter (17). It is also the largest by vertex count (66).

There are 37 configurations where QTM radius is the same as QTM diameter. The largest of these by diameter is 0x10F4018C7C0C63 (QTM diameter = 5), which seems like it can barely scramble. Another one that jumps out at me is 0x30000080200C11 (39 QTM diameter, 30 QTM radius). Among the puzzles where QTM radius is close to QTM diameter, it has an unusually high diameter.

Comparing graphs with and without considering orientation

Change in vertex count

When we stop considering orientation, the graphs for most configurations get smaller. The factor they shrink by roughly represents how many rotated/mirrored copies of each state we expect to find in the graph.

This quotient has an lower bound of 1 - if a cube has no self-similar states then its graph will have the same number of vertices whether or not we consider orientation. The upper bound is 48, since the cube has 24 orientations and each of those has a mirror image.

Large change

Exactly one puzzle has 48x as many vertices when orientation is considered - 0x28A00000414. When considering orientation it can reach 2544 configurations, and when ignoring orientation it can only reach 53 configurations.

Achieving this bound is quite impressive. It means that for any reachable configuration of this puzzle, all 48 orientations/mirrors are reachable and distinct from one another.

A few other puzzle get quite close to a 48x reduction. The largest of these by diameter is 0x20088000000C06 (27 QTM when considering orientation, 16 QTM when ignoring it). The largest of these by vertex count is 0x28000000C14 (7344 when considering orientation, 157 when ignoring it). This cube is also notable for having the largest vertex count if we limit outself to cubes with 0 modified center pieces (9528 when considering orientation, 206 when ignoring it).

No change

Nearly 1000 puzzles have the same number of vertices whether or not orientation is considered.

The largest by diameter are 0x108430C40007A3 (seen above), then 0x500180800C3D (seen above). The largest by vertex count are 0x10840084018423 (12647) and 0x100400B4200631 (11589).

Large graphs despite large changes

Some configurations still have comparatively large graphs, even after losing a big portion of their vertices when we ignored orientation.

0x1003DE80000403 (now 18th largest) lost 3/4th of its vertices. 0x3FE0010002D (now 60th largest), lost ~7/8th of its vertices.

Change in diameter

Large change

Most of the graphs with large percentage drops in diameter end up with ~2 vertices and don’t seem too interesting to me. A few that do seem interesting are 0x10525A02800C06 (14 QTM -> 4 QTM) and 0x300000B40D85A3 (14 QTM -> 2 QTM), Both drop from 5 HTM to 1 HTM. Another notable drop is from 0x53A40280007F (84 QTM -> 34 QTM).

Small change

There are some interesting examples of graphs whose vertex count drops significantly but whose diameter does not change.

A few standouts are 0x10840000000C03 (904->127 vertices, 23 diameter QTM), 0x3F6000005AD (2304->576 vertices, 49 HTM diameter).

Bonus - other stuff that seemed interesting

Banning “normal” corners

If we limit ourselves to puzzles where every corner is bandaged to an edge, we still have 81 puzzles to look at. If we care about orientation, 0x39e00100414 has the largest diamter (36 QTM) and 0x2da00000c14 has the largest vertex count (384). If we stop caring about orientation then 0x21088300000883 has the largest diameter (30 QTM) and the largest vertex count (257).

Banning “normal” edges

If we limit ourselves to puzzles where every edge is bandaged to another piece, we still have 132 puzzles to look at. Of these, 0x21081308440862 has the largest diameter (28 QTM) whether or not we care about orientation. 0x3fe0000041c has the largest vertex count (1092) when considering orientations and 0x1003fe8000043d has the largest vertex count (321) when ignoring orientations.

Banning “normal” corners and edges

If we restrict ourselves to puzzles with no “normal” corners and no “normal” edges, there are still 12 distinct puzzles to look at. These puzzles are all extremely constrainted but some seem like they can still meaningfully scramble.

0x3FE00000C1C has the largest vertex count (156) if we care consider orientation. 0x21189308C00C46 has the largest vertex count (25) if we do not consider about orientation. It also has the largest diameter whether or not we care about orientation. 0x83FE04000631 is in third place for vertex count whether or not we care about orientation.

Banning “abnormal” centers

If we restrict ourselves to puzzles where all centers are “normal” meaning no bandaging to the core or to edges, we have 26 puzzles to look at.

0x20088000000C06 (seen above) has the largest diameter (27 QTM) of these puzzles if we care about orientation. This was was noted above for have losing a large fraction of vertices when we start caring about orientation. 0x20088000000C03 has the largest diamter (20 QTM) if we do not care about orientation.

Fun graph shapes

Here are a bunch of puzzles where the shape of their graph seemed interesting to me:

0x210B0118C00C5A has two identical looking subgraphs connected by a forced chain. I wonder if the solve will require navigating the chain frequently - or even at all.

0x1084018C7C0C30 is imilar to the above but with a much longer chain connecting the segments.

0x1080018FFC0C21 has a very unusual graph. It has a few big loops that are all connected to each other, plus so not-too-useful looking long dead ends.

0x104000B5BDB5AD has very small graph with only a few nontrivial loops. It seems like it can scramble pretty well though and would make for a challenging and highly constrained solve.

0x30018080002D has an intersting graph shape. It is sparse but hard to untangle.

Closing thoughts (again)

I’d like to re-emphasize that diameter is not a good way to predict difficulty. I’m currently fighting with 0xC00B40D85A0 which a QTM diameter of 3 and am finding it pretty difficult. That said, I think it’s difficulty is quite different from the high-diameter puzzles discussed in those post.

I’m probably done looking at the diameter data for now, but I’m certainly not done analyzing bandaged cubes. One idea I had was a web tool to help find paths between different configurations. Another idea I’m considering is building a tool that finds non-trivial loops in a puzzle’s state space; for the purpose of finding useful algorithms. If I do either of these I’ll be sure to write a blog post too.

Written on December 20, 2022