# The 20MM (20 Move Method)

#### whauk

##### Member
Well, that's essentially what in trying to do with permutation for corners using stationary points at the moment.

Basically, I'm grouping scrambles according to stationary points and some other factors (like what cycle the other corners are in) and trying to find patterns in the generators and matrices but I can't spend much time doing it atm cause of college.
The last days I have read "stationary points" very often in this forum. Is it something else than just fixed points of the permutation? Because the notion of stationary points is usually used in the context of differentiable functions, whereas fixed point is a term for arbitrary automorphisms.

##### Member
The last days I have read "stationary points" very often in this forum. Is it something else than just fixed points of the permutation? Because the notion of stationary points is usually used in the context of differentiable functions, whereas fixed point is a term for arbitrary automorphisms.
In the context of matrices and transformations on general, it refers to any set of points which aren't affected by the transformation. For example, the line y=x is a set of stationary points for the transformation x=y

#### mark49152

##### Super Moderator
Staff member
Let me set aside the temptation to write this off as unrealistic, for a moment, and just throw some thoughts out.

Let's suppose the key to finding human solutions that are close to optimal is in avoiding the need to break the solve into phases, whether those phases are to solve subsets of pieces or to reduce to certain groups.

What would it take to have a single-phase solution?

I would guess there would have to be a metric of some sort that would measure how far the current state of the cube is from being solved. Obvious candidate metrics would be number of pieces solved, number of blocks formed, number of pieces oriented/permuted, etc.

Next, that metric has to satisfy some criteria to be useful for human solving. (1) It has to be possible to calculate the metric quickly enough for a given state, whether for speedsolving or FMC. (2) It has to be possible and practical to see from the current state which moves are preferred for reaching a next state with improved metric. And (3) there must be a linear progression through improving metric values from scrambled to solved - in other words, if it may be necessary to regress by making a move to a state with a worse metric value in order to make bigger jumps forward with subsequent moves, then the search space becomes impractical.

Obviously, metrics like number of pieces solved, number of blocks formed, number of pieces oriented/permuted might satisfy the first two criteria but cannot satisfy the third. And optimal solution length is a metric that would satisfy the third criteria but not the first two.

I have no idea if a suitable metric exists, although my guess would be that the best we could expect would be something workable for FMC, perhaps with some compromises, but not for speedsolving. To my mind it's the third criteria above that really makes it hard to find a suitable metric.

#### SpeedCubeReview

##### Member
This is just to see if it can be done, in the future it may become a speedcubing method, but i honestly believe that cfop/roux will be faster. It would be very helpful for FMC however. And i don't want to be rude, but i made this thread for people interested in helping, not people interested in telling me it can't be done, so if you don't believe it can be done, fine, but please keep it to yourself. Just a heads up for anybody who doesn't know, you can send me a contact request on Skype by clicking on the Skype icon under my name, and choosing send a contact request.
Welcome to the Forums. Since this is a thread for anyone to post on people can post what they want (As long as it passes the moderators).

I think people telling you that there isn't a method like you would find with CFOP is actually people trying to help. It may seem like they are putting you down but really its to help you on your quest in a more direct way.

##### Member
Welcome to the Forums. Since this is a thread for anyone to post on people can post what they want (As long as it passes the moderators).

I think people telling you that there isn't a method like you would find with CFOP is actually people trying to help. It may seem like they are putting you down but really its to help you on your quest in a more direct way.
I think that's he's more refering to posts which say "it can't be done. Stop trying and wasting your time". And I think if people read the thread they would realise that he already said the method won't be like CFOP.

##### Member
I have no idea if a suitable metric exists, although my guess would be that the best we could expect would be something workable for FMC, perhaps with some compromises, but not for speedsolving. To my mind it's the third criteria above that really makes it hard to find a suitable metric.
This would be tough. Even the obvious metrics that you mentioned don't seem suitable. Compare a cube (cube A) that's one T-Perm away from being solved vs a cube (cube B) that's F U R away from being solved:
1) Cube A has 4 unsolved pieces, cube B has 16.
2) All pieces are oriented on cube A, on cube B there are 4 to 2 miss-oriented pieces depending on orientation.
3) This one I'm not too sure about, depends on how you define a block.

The one thing that I can see looks better on cube B than on cube A would be something like "amount of moves that can be performed without decreasing the amount of formed blocks in the cube".

#### Lucas Garron

##### Moderator
Staff member
Next, that metric has to satisfy some criteria to be useful for human solving. (1) It has to be possible to calculate the metric quickly enough for a given state, whether for speedsolving or FMC. (2) It has to be possible and practical to see from the current state which moves are preferred for reaching a next state with improved metric. And (3) there must be a linear progression through improving metric values from scrambled to solved - in other words, if it may be necessary to regress by making a move to a state with a worse metric value in order to make bigger jumps forward with subsequent moves, then the search space becomes impractical.
If you aren't familiar with (1) pruning tables or (2) IDA*, I highly recommend you take a look at them before going further.

Searching with heuristics is a fairly well-understood subject.
Presumably, if it was easy to find good heuristic that satisfies (3), we would have used it to write puzzle solving programs that generate solutions with almost no work. But so far, no one has found systematic heuristics for a cube that give near-optimal solutions without a lot of backtracking.

For puzzles like the 3x3x3 cube, the best we know how to do is use a few tricks to bring brute force down to the point where computers happen to be able search quickly enough. (But even that breaks down quickly for 4x4x4.)

That doesn't mean that there isn't progress to be made, or that you can't learn from trying. But if you want to make progress in this field, you should at least know the state of the art among algorithms that take your approach.

##### Member
If you aren't familiar with (1) pruning tables or (2) IDA*, I highly recommend you take a look at them before going further.
Knowing about IDA would definitely help someone come up with an algorithm to solve the cube close to optimally. However, I'm having trouble imagining how knowledge of pruning tables helps when trying to come up with an algorithm that's intended for a human to perform in a limited period of time.

FMC is still in its infancy, you can learn most of what's been discovered from a single PDF and achieve world class status much faster than in any other event I can think of. In other words, there's A LOT of progress to be made. However, it's over ambitious to try to discover a method that consistently produces 20 move (or less) solves (especially if your only insight so far is that "it is a variable method, meaning the method may change"), I'd be happy with one that could get you sub-30 move solves on most scrambles.

#### mark49152

##### Super Moderator
Staff member
Presumably, if it was easy to find good heuristic that satisfies (3), we would have used it to write puzzle solving programs that generate solutions with almost no work. But so far, no one has found systematic heuristics for a cube that give near-optimal solutions without a lot of backtracking.
Yes of course. The point of my post was to speculate that finding a metric (heuristic) that satisfies (3) is the key to near-optimal human solving. I certainly didn't claim it was easy; quite the opposite. Furthermore, it would also have to satisfy (1) and (2), which wouldn't be necessary for a computer solution.

By the way, I'm not working on this, and don't intend to. I was just throwing out some thoughts.

#### cuBerBruce

##### Member
I am going to try to put what the OP is claiming to try to do into a mathematical framework.

Let S be the set of positions that can be solved in 10 or fewer moves. (The value of 10 here is rather arbitrary, but I'll just use this value for this discussion.)

Let G be some subset of the elements of S. (I chose G for "goal". The goal of the first phase is to reach an element of G.) These elements may be chosen based upon some properties of the positions, such as some relationship having to do with the permutation and orientation of the pieces.

Let G[sub]N[/sub] be the set of elements in G that are optimally solved in N moves. (That is, we partition the set G based upon the number of moves required to solve each position.)

We want to find a "method" such that we can show that for every position of the cube, we can reach some position in G[sub]N[/sub], for some N, using no more than 20-N moves.

(And not only this, but we also want this method to be human learnable.)

There are two big obstacles with this.

1. Choosing a set G small enough to be human learnable (the human must know how to optimally solve every element of G). But G must also be large enough to make it possible to accomplish the first phase in few enough moves.

2. Figuring out how to reach an element of G from an arbitrary position in few enough moves. If we ignore the "human learnable" requirement, we "simply" need to show that this is possible.

I think trying to make (2) a manageable size problem is generally going to force increasing the number of elements in G, making the first obstacle worse. Likewise, trying to make the first obstacle manageable is going to tend to make the 2nd obstacle all the more difficult, if not impossible.

In the end, I assume some concessions will need to be made. These could include:

1. Reduce the number of positions that must be solved in 20 moves. If, say, we only require that only 99% of the positions need to be solved in 20 moves, we remove the requirement that all of the millions of 20f* positions need to be solved optimally.

2. Require the average move count to be 20, rather than 20 as the worst case.

3. Relax the move count that is required to be achieved to some larger number (21 or 22 or 23, etc.).