# Move count metrics for big cubes - standards and preferences

#### Cride5

I've been looking for a 'definitive' source on the metrics people are using to measure move counts for big cube algorithms. Unfortunately I couldn't find anything on the Wiki, and information within threads of this forum was pretty sparse to say the least.

This post by qq here seems to cover the range of possible metrics pretty well:
.. but I'm not sure how standardised this sort of system is.

Just as an initial proposition (based on qq's idea) how does this sound:
* STM - Slice [half] Turn Metric - Same meaning as 3x3, counts half/quarter turns of a single slice (layer) as 1 move.
* SQTM - Slice Quarter Turn Metric - As above, but half turns count as two.
* WTM - Wide [half] Turn Metric - A turn of any number of contiguous layers including exactly one outer layer, by the same angle counts as one move.
* WQTM - Wide Quarter Turn Metric - As above, but half turns count as two.
* BTM - Block [half] Turn Metric - A turn of any number of contiguous layers by the same angle counts as one move.
* BQTM - Block Quarter Turn Metric - As above, but half turns count as two.
* ATM - Axial Turn Metric - Any number of turns on one axis count as one move, for example L r2 = 1 atm.

Another possible metric could exist, something between MTM and ATM. It would be like MTM but relaxing the constraints on the slices being contiguous, but requires that each slice is turned by the same angle. Although it's possible I don't think this sort of metric would be useful. What do others think, would this be useful, and for what sort of applications?

So basically, I'm looking for your opinions on creating some sort of standard for bigcube turn metrics, specifically:
* Do the metrics above make sense?
* Do they cover the range of possible applications?
* Are the names appropriate?
* What should we use as the 'standard' metric from a speedsolving perspective - BTM seems to be preferred, but is it the best at approximating algorithm execution time?

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#### Lucas Garron

##### Member
Good idea; I intend to compile standards and start using them, but I'm reluctant to start on stuff like this.

* MTM - Multislice [half] Turn Metric - A turn of any number of contiguous layers by the same angle counts as one move.
* MQTM - Multislice Quarter Turn Metric - As above, but half turns count as two.
I suggest BSTM and BQSTM.

Does this also remind you of complexity classes?

#### rokicki

##### Member
As I start looking at the 4x4x4, I'm definitely planning to focus on BTM, so if this doesn't match what most people think as the most "interesting" metric for this size, please let me know. I think BTM is clearly the most natural metric.

#### Lucas Garron

##### Member
As I start looking at the 4x4x4, I'm definitely planning to focus on BTM, so if this doesn't match what most people think as the most "interesting" metric for this size, please let me know. I think BTM is clearly the most natural metric.
It's the most interesting because that's how you turn a physical cube most of the time.
Real slice turns –where only an inner slice moves– are possible on 4x4x4 (I use them for r2), but uncommon on big cubes in general.

#### Cride5

@lucas - yup, in the context of using these different metrics in a solver I can see how it sort of changes the definition/complexity of the problem, mainly by changing the branching factor.

With the choice of acronym I went for the shortest sequence of letters to describe the metric, without clashing with any other cubing related acronyms. BSTM is good from a semantics point of view, but perhaps a little too verbose?

@ rokicki + lucas ... I would deffo agree that BTM is probably best as it most accurately represents the individual turns I make when solving 4x4 with Reduction. I wonder whether K4, Roux by 4, or Sandwich users would agree tho

#### irontwig

##### Member
I guess that execution speed can probably be approximated pretty good with a (weighted) average between SQTM and BQTM.

#### Matt S

##### Member
For approximating my own speed on algs for very large cubes, I would use a combination of BTM and ATM, with the first turn on each axis counting double to account for aligning the cube.

From a theoretical perspective, I'm fond of SQTM for its mathematical properties and BTM for its physical properties.

#### cuBerBruce

##### Member
Just as an initial proposition (based on qq's idea) how does this sound:
* STM - Slice [half] Turn Metric - Same meaning as 3x3, counts half/quarter turns of a single slice (layer) as 1 move.
* SQTM - Slice Quarter Turn Metric - As above, but half turns count as two.
* BTM - Block [half] Turn Metric - A turn of any number of contiguous layers including exactly one outer layer, by the same angle counts as one move.
* BQTM - Block Quarter Turn Metric - As above, but half turns count as two.
* MTM - Multislice [half] Turn Metric - A turn of any number of contiguous layers by the same angle counts as one move.
* MQTM - Multislice Quarter Turn Metric - As above, but half turns count as two.
* ATM - Axial Turn Metric - Any number of turns on one axis count as one move, for example L r2 = 1 atm.
It seems like what to call various metrics on big cubes has never come to much of any agreement by people. Hence we don't really have much in the way of strong de facto standards.

I have been using "block turns" to mean moving any contiguous block of layers with respect to the rest of the cube. I am surprised it's being proposed to mean something else here. The proposal here is to use "block turns" for what was usually referred to as twists on the old cube-lovers email list. Some people have called it face turn metric (as it is always involves turning at least a face layer). The WCA regulations already defines the same metric has "half-turn metric." I really think BTM (block turn metric) is a misleading name.

I suggest BSTM and BQSTM.
I think "QS" seems to imply you're allowed to move a quarter of a slice, which of course is ridiculous. I think an S between the Q and the T is bad.

#### mrCage

##### Member
BTM all the way. It's all i care about when making pattern sequences for instance. BQTM seems artificial to me:tu

Per

#### Kenneth

##### Not Alot
I use moves like r l at times and with practice it is doable in one go, does that only fit in the axis group or is it like I think, a antislice move?

#### mrCage

##### Member
I use moves like r l at times and with practice it is doable in one go, does that only fit in the axis group or is it like I think, a antislice move?
Yes, that is an "inner antislice" move. It would be 1 turn in ATM. 2 turns in BTM. No one would consider antislice moves as 1 turn on 3x3x3 so why do that on bigger cubes? Be it outer (pure) or inner ones. On cubes larger than 6x6x6 one could even have inner block antislice turns... Oh well

Per

#### irontwig

##### Member
No one would consider antislice moves as 1 turn on 3x3x3
Tony Snyder?

#### Stefan

##### Member
I thought BTM is for inner blocks as well. I like "PTM" (plane turn metric) for turing along just one plane, i.e., any number of contiguous layers including an outer layer. That name is more descriptive and thus clearer, I think.

#### cmhardw

No one would consider antislice moves as 1 turn on 3x3x3
Tony Snyder?

Though I'm not sure if she would actually count as one move, Jessica Fridrich provides a notation that presents anti-slice turns as one unit, so possibly counted as one turn in some form of Anti-slice metric.

[/hijack]

Chris

#### Lucas Garron

##### Member
I thought BTM is for inner blocks as well. I like "PTM" (plane turn metric) for turing along just one plane, i.e., any number of contiguous layers including an outer layer. That name is more descriptive and thus clearer, I think.
I can see that interpreted as "turn a single plane" where "plane" becomes a synonym for "slice."

#### qqwref

##### Member
I've always used BTM (block turn metric) to mean turning any consecutive group of slices on one axis by the same (directed) angle. I don't see any reason to change this convention now. I feel like this is the most natural metric to use because it doesn't restrict too far (pretty much anything that feels like a single move is) but also doesn't allow anything crazy to be counted as a single move (like, for instance, Axial would).

PS: For the metric where any block turn which includes an outer layer counts as one move, I suggest MTM (multislice turn metric, because of the multislice moves on cube simulators), WTM (wide-turn metric), or FTM (face turn metric).

#### Lucas Garron

##### Member
WTM (wide-turn metric)
I like this. Since SiGN seems to have lost quite a bit, establishing moves like Rw and 3Rw as "best-practice" is a good idea since it's unambiguous. Calling them all "wide turns" and using "WTM" is maybe not the most intuitive, but at least consistent.

#### qqwref

##### Member
I don't like the idea of writing Rw2 for a single turn (and then Rw2' and Rw22). Why not 2Rw?

#### Christopher Mowla

Because it looks odd that way. Only those who are accustomed to previous notations will need to make that minor adjustment. But, I guess it could be more consistent with the inner-slice turns for bigger cubes (e.g. 2f2 is in the second portion of the 7X7X7 notation) by writing 2Rw2 instead of Rw22.

Hence, it could be
$$\underline{\text{number of quarter turns}}\left( \text{R,L,U,F,D,B} \right)\text{w}\underline{\left( \text{ }\!\!\#\!\!\text{ of consecutive slices} \right)}$$

E.G. 2(R)w2 =2Rw2

Edit: Maybe the reason I prefer Rw22 more instead of 2Rw2 is because I used my notation in all of my research documents on odd parity algorithms (several hundred typed pages in all). Typing out Rw22 is easier on the fingers than 2Rw2, in my opinion.

Edit: Also, I prefer not to write Rw (like WCA notation) because that is only exception to the overall plane turns on big cube sizes in general (obviously). So, that's why I include the '2' there to indicate that I am moving two slices simultaneously, even though in WCA notation it is not so.

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#### TMOY

##### Member
FYI, in the very first competitions where 6^3 and 7^3 were held as unofficial events, we used R_2 (with 2 as subscript) for a double layer quarter-turn, and R2 as usual for a single layer half-turn. It was really confusing and led to cubes being almost never correctly scrambled, and that's why the 2 was moved to the beginning when the events were made official.