# Average Square-1 shape distances, for all shapes



## xyzzy (Jul 30, 2021)

Not sure if this really deserves a thread on its own, but I guess a mod can merge it into the squan discussion thread if not.

The left/right labelling of the paws and fists follows Jaap's square-1 page.
Peacock = cccccee (aka "pair")
Arrow = ccccece (aka "perpendicular")
Badge = cccecce (aka "parallel", "crown")
The numerical shape names should be self-explanatory (the number of edges between the corners, read clockwise, skip all zeros).

This table lists the average optimal slices to go from a random shape (weighted by frequency) to the specified shape. For the "exact" column, exactly that shape has to be the result (e.g. for the first row, 24/arrow doesn't count as the correct shape; rotating is not allowed), while the "equiv" column allows for all symmetrically equivalent shapes (e.g. for the first row, these are arrow/24, arrow/42, 24/arrow, 42/arrow).


shapeexactequivarrow/24​2.0310​1.8515​r.fist/scallop​2.0962​1.8885​peacock/123​2.1077​1.9228​r.fist/shield​2.2289​1.9516​arrow/123​2.2181​2.0288​arrow/15​2.1631​2.0305​peacock/15​2.1528​2.0321​peacock/114​2.0593​2.0332​badge/123​2.3061​2.0517​kite/l.paw​2.3583​2.1060​l.fist/l.paw​2.2811​2.1071​arrow/33​2.1354​2.1223​shield/l.paw​2.3714​2.1321​barrel/l.paw​2.4171​2.1756​r.fist/l.paw​2.3464​2.2213​mushroom/r.fist​2.5731​2.2273​badge/15​2.4812​2.2365​l.paw/scallop​2.4002​2.2567​arrow/114​2.3138​2.3051​peacock/24​2.5068​2.3219​mushroom/l.paw​2.5759​2.3322​kite/scallop​2.3605​2.3475​r.fist/barrel​2.6993​2.3491​peacock/33​2.3681​2.3583​arrow/6​2.3719​2.3654​badge/114​2.3888​2.3801​r.paw/l.paw​2.4296​2.4252​l.paw/l.paw​2.6357​2.4747​kite/shield​2.5394​2.5329​badge/24​2.6656​2.5895​arrow/222​2.5965​2.5922​kite/r.fist​2.9951​2.6580​square/scallop​2.6911​2.6846​square/l.paw​2.8874​2.7047​square/shield​2.7765​2.7700​badge/6​2.7792​2.7760​kite/mushroom​2.8233​2.8189​mushroom/scallop​2.8548​2.8483​barrel/scallop​2.8559​2.8494​mushroom/shield​2.8668​2.8603​badge/33​2.8858​2.8825​r.fist/r.fist​2.9685​2.8989​kite/barrel​2.9092​2.9048​l.fist/r.fist​2.9244​2.9146​peacock/6​2.9701​2.9603​square/r.fist​3.2833​2.9625​barrel/shield​2.9869​2.9804​square/mushroom​2.9962​2.9918​scallop/scallop​3.0033​3.0033​star/53​3.0147​3.0103​star/62​3.0174​3.0131​shield/scallop​3.0381​3.0283​star/44​3.0587​3.0560​badge/222​3.0756​3.0734​shield/shield​3.1294​3.1294​star/71​3.1697​3.1653​peacock/222​3.1854​3.1822​square/barrel​3.2811​3.2768​mushroom/barrel​3.2860​3.2817​star/8​3.2942​3.2898​kite/kite​3.6264​3.6264​mushroom/mushroom​3.7526​3.7526​barrel/barrel​3.7531​3.7531​kite/square​3.8923​3.8902​square/square​4.6243​4.6243​

(Is there a way to make the table not extend to the edges of the post? Could've sworn there was an option for that in the past.)

---

Interesting to see that square-square (i.e. solving cubeshape as per normal) is actually the worst shape to aim for. (Which makes sense in hindsight, because it's only connected to kite-kite.)

Among the depth-3 cubeshape cases, scallop-kite has the lowest average distance (2.35 slices), opposite paws has the second lowest (2.43 slices), and everything else has 2.77 slices or more. This provides some justification for why scallop-kite is such a good beginner/intermediate-level method for solving cubeshape: it's Pareto-optimal between being easy to get into, and also being easy to get to cubeshape from.

Lining it up against the other Pareto-optimal cubeshape methods making use of an intermediate shape:
arrow-24 (1.85 slices to this, 5 more slices to cubeshape)
arrow-123 (2.03 slices to this, 4 more slices to cubeshape)
scallop-kite (2.35 slices to this, 3 more slices to cubeshape)
opposite fists (2.91 slices to this, 2 more slices to cubeshape)
square-square (4.62 slices to this, 0 more slices to cubeshape)

(There is no Pareto-optimal method that uses a depth-6 or depth-7 intermediate shape because those methods are uniformly worse than arrow/24. Using kite-kite as an intermediate shape is just solving cubeshape normally.)


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## Lucas Garron (Jul 30, 2021)

Oooh, very neat!

I'm curious, if you plot the distances between two shapes as a 2D heat map, does the average heat in a row/column give a good intuition for these values?
What about a graph where node sizes represent these average distances?


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## xyzzy (Sep 30, 2021)

What if you could solve cubeshape and OBL together? Think VLS, but for square-1: you get the shape into a specific configuration, then simultaneously solve OBL while finishing off cubeshape.

This method has obvious downsides, but we'll get to that later. The more interesting question is: _which_ specific configuration should we choose? Just like the table in the initial post, and like the table I posted in the Yoyleberry thread, with the help of a Computer we can quite easily determine the average distance (in STM) for each possible choice of intermediate shape.

(Technical note: By "OBL", I mean orienting the pieces wrt the gripped equator piece (the same one used for scrambling). There's a slightly more general version of OBL where you can choose the other equator piece, and an even more general version where you allow swapping the top and bottom layers altogether. I'm not using those. This should only affect the slice count of states that are already in cubeshape. For the same reason, we can still safely consider mirrored shapes to be equivalent.)


chosen shapeto this shapeto CS+OBLsuml.fist/r.fist​2.9146​5.1192​8.0338​l.fist/l.paw​2.1071​5.9678​8.0749​square/shield​2.7700​5.3616​8.1316​arrow/123​2.0288​6.1837​8.2125​square/mushroom​2.9918​5.3633​8.3551​kite/scallop​2.3475​6.0857​8.4332​kite/kite​3.6264​4.8220​8.4485​peacock/114​2.0332​6.4367​8.4699​r.paw/l.paw​2.4252​6.0494​8.4746​square/square​4.6243​3.9433​8.5675​arrow/24​1.8515​6.7257​8.5773​peacock/123​1.9228​6.6853​8.6081​



Spoiler: full table




chosen shapeto this shapeto CS+OBLsuml.fist/r.fist​2.9146​5.1192​8.0338​l.fist/l.paw​2.1071​5.9678​8.0749​square/shield​2.7700​5.3616​8.1316​arrow/123​2.0288​6.1837​8.2125​square/mushroom​2.9918​5.3633​8.3551​kite/scallop​2.3475​6.0857​8.4332​kite/kite​3.6264​4.8220​8.4485​peacock/114​2.0332​6.4367​8.4699​r.paw/l.paw​2.4252​6.0494​8.4746​square/square​4.6243​3.9433​8.5675​arrow/24​1.8515​6.7257​8.5773​peacock/123​1.9228​6.6853​8.6081​kite/barrel​2.9048​5.7376​8.6424​arrow/15​2.0305​6.6506​8.6811​r.fist/scallop​1.8885​6.8584​8.7469​peacock/24​2.3219​6.4392​8.7611​arrow/33​2.1223​6.6502​8.7726​arrow/114​2.3051​6.4898​8.7949​peacock/15​2.0321​6.8216​8.8537​badge/123​2.0517​6.8135​8.8651​r.fist/shield​1.9516​7.0498​9.0014​kite/l.paw​2.1060​6.9033​9.0093​shield/shield​3.1294​5.8955​9.0249​barrel/scallop​2.8494​6.2102​9.0596​r.fist/l.paw​2.2213​6.8751​9.0964​scallop/scallop​3.0033​6.1037​9.1069​mushroom/shield​2.8603​6.2841​9.1443​peacock/33​2.3583​6.7898​9.1481​barrel/barrel​3.7531​5.3959​9.1490​badge/15​2.2365​6.9404​9.1769​barrel/l.paw​2.1756​7.0143​9.1899​shield/l.paw​2.1321​7.0784​9.2105​peacock/222​3.1822​6.0857​9.2679​square/l.paw​2.7047​6.5682​9.2729​l.paw/scallop​2.2567​7.0571​9.3138​star/44​3.0560​6.2800​9.3360​arrow/6​2.3654​6.9841​9.3495​star/62​3.0131​6.3445​9.3575​square/scallop​2.6846​6.6776​9.3622​kite/shield​2.5329​6.8400​9.3729​mushroom/l.paw​2.3322​7.0620​9.3943​badge/24​2.5895​6.8053​9.3948​peacock/6​2.9603​6.4527​9.4130​badge/114​2.3801​7.0633​9.4434​l.paw/l.paw​2.4747​6.9780​9.4527​arrow/222​2.5922​7.0457​9.6379​star/8​3.2898​6.3535​9.6433​shield/scallop​3.0283​6.6343​9.6626​mushroom/r.fist​2.2273​7.4702​9.6975​barrel/shield​2.9804​6.7380​9.7184​mushroom/mushroom​3.7526​5.9722​9.7248​r.fist/barrel​2.3491​7.4510​9.8001​badge/6​2.7760​7.1180​9.8939​badge/33​2.8825​7.0576​9.9401​mushroom/scallop​2.8483​7.1261​9.9744​kite/r.fist​2.6580​7.4751​10.1331​kite/mushroom​2.8189​7.3241​10.1430​r.fist/r.fist​2.8989​7.2629​10.1617​star/53​3.0103​7.2976​10.3079​badge/222​3.0734​7.2449​10.3183​square/r.fist​2.9625​7.3951​10.3576​star/71​3.1653​7.2180​10.3833​mushroom/barrel​3.2817​7.2469​10.5286​square/barrel​3.2768​7.4514​10.7282​kite/square​3.8902​7.8318​11.7220​




Unsurprisingly, the best shape to do OBL from is square/square _if_ you're already in that shape. If you include the slices needed to get to the chosen shape, though, good fists ("l.fist/r.fist" in the above table) wins: it's over half a slice move better than solving cubeshape then doing OBL normally! In fact, normal cubeshape-then-OBL is tenth place in this list.

Now, the drawbacks:
1. This doesn't work with CSP.
2. Solving OBL from any asymmetrical shape will require 4900 algs. You can allow swapping the top/bottom layers to cut this by half (just insert a (6,6) before the last slice), but that's still 2450 algs.

It's absolutely, 100% not viable for actual speedsolving. You can incorporate CSP into this, but that would double the already huge number of algs, in addition to making inspection even harder.

On another note, cubeshape+OBL from any shape can always be done in at most 8 slices (average optimal 6.7131); CSP+OBL from any shape can always be done in at most 9 slices (average optimal 7.0413).


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## Thom S. (Oct 1, 2021)

xyzzy said:


> 1. This doesn't work with CSP.


What if you do CSP up to Good Fist, then CS+OBL, making sure thst every Algorithm doesn't disturb Parity?
Not that it makes this any more viable.


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## xyzzy (Oct 1, 2021)

Thom S. said:


> What if you do CSP up to Good Fist, then CS+OBL, making sure thst every Algorithm doesn't disturb Parity?
> Not that it makes this any more viable.


That could work in theory.

My wild guess is that most of the good-fists OBL solutions are just going into cubeshape in one of two ways (this won't affect parity), then doing the normal OBL alg from there, so perhaps it won't "truly" need 2450 algs; you just need to know which of two ways to finish off cubeshape to use. (And maybe it's partially intuitive like normal OBL is?)

Spending this much effort to shave off half a move is pretty silly regardless.


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