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 Post subject: Number of positions vs. number of states
PostPosted: Tue Jul 16, 2013 5:54 pm 
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I've heard of the total count of permutations of some puzzles described as either being the total number of "possible positions" or "possible states." My question is, are these definitions actually different? I read somewhere that the Square-2 has a total of 1,240,896,803,466,478,878,720,000 possible positions but only 8,617,338,912,961,658,880,000 possible states. So, do these terms mean different things? And what exactly do they mean?

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 Post subject: Re: Number of positions vs. number of states
PostPosted: Tue Jul 16, 2013 6:11 pm 
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I don't think there is an official (agreed upon) term. "positions" versus "states" seem like synonyms in the context of counting unique permutations on a puzzle. I'm sure I've used both.

One possible source of confusion when you see two different numbers probably stems from counting as though the puzzle has "super stickers" versus actually visually distinct pieces. For example, on the Rubik's cube you can't see the rotation of the centers which some people count as distinct states even though visually they are not. The other sources of numerical differences could come from duplicate positions due to re-orientation of the puzzle or even the total number of positions if you allow the puzzle to be taken apart and put back together.

I've even seen the word "symmetries" used by a mathematician (Marcus du Sautoy) to describe the total number of states you can assemble the puzzle into. This seems like a blatant misuse of the word though.

Jaap uses the term "positions" and I think his "vote" counts for a lot. I find myself using the term "states" a lot due to the mental model(s) I use for thinking about twisty puzzle.

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 Post subject: Re: Number of positions vs. number of states
PostPosted: Wed Jul 17, 2013 3:10 am 
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bmenrigh wrote:
...Jaap uses the term "positions" and I think his "vote" counts for a lot. I find myself using the term "states" a lot due to the mental model(s) I use for thinking about twisty puzzle.
I try to avoid the term "permutations" because mathematicians use it in a more strict way (Permutations of the elements of a group) that doesn't include per se - at least for me - "orientation". See here:
Allagem wrote:
I've played with a simulated version of this puzzle for about an hour and I am quite convinced that the pieces can only be rearranged into 336 permutations (not counting orientation...
I usually use "configurations" (or the synonym "states") to describe the visually distinguishable arrangements of puzzle pieces including different orientations.
"Positions" is a bit ambiguous in my view. E.g. I would hesitate to accept that a turned centre of a 3x3x3 Super Cube is at a different position. Certainly, it is in a different state or configuration.

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 Post subject: Re: Number of positions vs. number of states
PostPosted: Wed Jul 17, 2013 6:57 am 
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In general, is it not a universal problem that the terminology used when discussing a subject in a technical context differs from the terminology used by laymen when discussing the same subject, with this sometimes leading to confusion when these contexts mix? Puzzles in general are certaintly a subject for which mixing of technical and layman contexts is common.

Considering that many twisty puzzles are physical models of groups, it raises the question of what terminology would a Group theorist use?
-I am fairly sure permutation is the technical term for the number of ways in which n objects can be placed into n spaces.
-Does group theory even allow for elements can can have multiple orientations within a given position? If so, what would the group theory term for the collective orientation of a group of like elements?
-Am I correct that State would be the proper group theory term when taking permutation and orientation of all elements into consideration?
-I have heard the term orbit used to describe distinct sets of states that are isolated from each other under a group's operators, but both form a valid group with the given operators, but can only be bridged with illegal operators, but I do not know if this is how the term orbit is actually used in group theory.

Then again, is group theory well established enough to have an agreed upon terminology as I know many mathematical feilds are often independantly developed with each mathematician using their own terminology and it is only later that effort is made to standarize terminology for the sake of easing communication.

Also, I apoligize if any of what I said is difficult to understand. I understand the group theory is the underlying feild of mathematics relevant to twisty puzzles, but I know every little about group theory.

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 Post subject: Re: Number of positions vs. number of states
PostPosted: Wed Jul 17, 2013 7:09 am 
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I always thought of positions as the number of ways the pieces can be arranged via scrambling from solved, while the number of states is the number of ways the pieces can be arranged via assembly. Some states are not possible positions by this terminology, making the number of states always higher than positions for platonic twisty puzzles.

I'm probably pretty far off, but this is my understanding. It is a simpler definition in my opinion, but it could just be wrong.

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 Post subject: Re: Number of positions vs. number of states
PostPosted: Wed Jul 17, 2013 7:50 am 
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A permutation is a change of order of elements. This includes both change of position and/or change
of orientation. For example, if we wish to include the centers of a 3x3x3 cube, then we may simply
number each side of each center square and add this to the permutation cycles.

In general, a permutation may be used to express the connection between two different positions,
of the 3x3x3 cube, but unless it is expressed into the simple (4-cycle) moves of the cube (L,R,D,U,F,B),
it can be quite difficult to back-trace that route. In fact those six (4-cycle) moves are permutations
which "act" on the object (i.e. cube).

Regarding states versus positions, I may give an example of what I use based on the folding plate puzzles.
A puzzle with four plates has eight different flat positions, eight different "circle" positions, and eight different
"star" positions. But it has three states: flat, circle, and star.

So... leaving aside the 3x3x3 normal cube, and moving to a 3x3x3 Mixup Cube, we may have a cubic state,
as well as many other states with pieces pointing in or out (insert your state definitions here).

:)


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 Post subject: Re: Number of positions vs. number of states
PostPosted: Wed Jul 17, 2013 2:16 pm 
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I don't think group theory deals with orientations directly, but you can consider permuting stickers instead of permuting and orienting pieces. Then you only have permutations to deal with, which group theory certainly covers. This becomes more complicated for puzzles which cannot be described by group theory, such as the square-1.

More on topic, I personally use 'positions' and 'states' to mean the same thing. In regards to valid states and states possible with disassembly and reassembly, the other words in the sentence tend to make it clear what is being referring to in context and I don't think specific words for each is necessary, nor likely to catch on and be used consistently.


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 Post subject: Re: Number of positions vs. number of states
PostPosted: Wed Jul 17, 2013 3:24 pm 
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To echo what Matt said, piece orientation can be handled by considering the permutation of stickers.

I think using "permutation" is particularly problematic for other reasons though. It is much harder to define a permutation when a puzzle is bandaged. Take a 3x3x3 where you've glued one corner to one edge. That new macro-piece replaces two pieces when permuted. The only way to handle this sort of thing is to treat the permutation description of the puzzle as unbandaged (the bandaged pieces is 4 stickers) but where the bandaged parts move together.

Worse, jumbling really muddies the meaning of the word permutation. It's really hard to describe how stickers have been permuted when a puzzle is jumbled. It's much better to think of these sorts of cases as a "position" or "state".

Furthermore, when a puzzle has non-sticker-permutation properties like a locked axis (in the case of some of Oskar's new puzzles), it's especially hard to handle the state of the puzzle in terms of a permutation. In this case I like "state" a bit more than "position" because the word state can encompass things that are a stretch for the word "position" to handle.

Group theory doesn't help much here. When a puzzle forms a group, each state of the puzzle is one element in the group. Calling positions/states an "element" is probably more confusing than its worth. Plus, this also has trouble handling bandaging and jumbling and puzzles that don't form a group.

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