Reverse scrambles

Tim Reynolds

Hi puzzle theory people,

Lots of WCA competitions have adopted the policy of "use a reverse scramble on an extra attempt". I'm interested in whether this is, puzzle-theoretically, a good option. [note: in contrast to the inverse scramble, I'm talking about the reverse scramble. For instance, the reverse scramble of F R U is U R F, not U' R' F']

For now I'm just thinking of 3x3, but I guess we can extend it to other puzzles too.

-Are there any relationships between the regular scramble and the reverse scramble? I can see that permutation parity must be the same.

-Is there any statistical problem with the distribution of positions reached by reverse scrambles?

If it helps, we could assume that we're dealing with the first scramble that Cube Explorer stops on (usually 21 moves). I guess technically, if CubeExplorer is deterministic, then a person could memorize the cube during inspection, cause a timer malfunction, run to their computer and plug the cube into Cube Explorer, take the scramble it gives, and reverse it. Are there any other conceivable issues we can see with reverse scrambling?

Thanks,
Tim

qqwref

Member
Note that there is more than one possible reverse scramble for any given position, e.g.:
- U2 => U2
- R U R' F' U2 L' U' L F => R' U' R F U2 L U L' F', which is not the same as U2

So without knowing the original scramble, someone should not be able to determine the new position. I do think Cube Explorer is deterministic though, so assuming the same settings are used (including the number of times the green arrow was used to get a shorter solution) someone could find the new scramble. However, I think we can prevent this by just not letting people leave the table while the cube is being rescrambled (with high "priority", and even by the judge if necessary). Nobody can enter in an entire position and get the solution (at all, let alone fast enough to be able to preinspect and/or memorize a solution to the new scramble), especially without the judge noticing.

TMOY

Member
For BLD events, the inverse scramble is bad because it yields the same cycles (reversed) as the original scramble. The reverse scramble doesn't suffer from that problem.

ben1996123

Banned
For BLD events, the inverse scramble is bad because it yields the same cycles (reversed) as the original scramble. The reverse scramble doesn't suffer from that problem.
If it had the same cycles, wouldn't that make it fair though because it wouldn't mean that one person is getting a scramble that's much easier/harder than the one that everyone else is getting?

Tim Reynolds

If it had the same cycles, wouldn't that make it fair though because it wouldn't mean that one person is getting a scramble that's much easier/harder than the one that everyone else is getting?
The problem is that, if the competitor starts memorizing the forward scramble, realizes there was a timer malfunction, and then restarts on the inverse scramble, they start the solve already knowing part of the memorization.

Ranzha

Friendly, Neighbourhoodly
The problem is that, if the competitor starts memorizing the forward scramble, realizes there was a timer malfunction, and then restarts on the inverse scramble, they start the solve already knowing part of the memorization.
Having extra scrambles handy is another solution. Easy fix.
For blind, because of the nature of the memorisation, perhaps orienting the cube differently could fix the problem as well. The competitor should not be able to discern the translation.

Bob

Having extra scrambles handy is another solution.
Yes, of course. Here, I agree with you completely. However, we are trying to see if there is anything wrong with the other practice. We are mostly interested in 3x3 as some other puzzles will run into issues:

Square-1: Not usually possible to use reverse scramble.
Pyraminx: Same tip positions (well, inverted, but that's close enough)
Megaminx: Reverse scramble is scrambled in a different spirit than forward scrambles. Here I would think a reverse scramble should instead be the lines from bottom to top.

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BlackStahli

Member
Just a thought, but won't there be 4 possible scrambles if you have any one given scramble just be reversing the moves and the order of the moves?
For example: D' R F' D R U' R F2 U R can also give
D R' F D' R' U R' F2 U' R'
R' U' F2 R' U R' D' F R' D (which, if done after the original scramble, returns the cube to the solved state, and vice versa)
R U F2 R U' R D F' R D'
and I can assume that all 4 sequences of moves result in different scrambles. (ignore the length of the "scramble", just wanted to make an example)
EDIT: I can't really tell how "much" the scrambles are different from all aspects (since I can't bld) but from the perspective of a regular 3x3 solver I can say that they are all different.

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TMOY

Member
The problem is that, if the competitor starts memorizing the forward scramble, realizes there was a timer malfunction, and then restarts on the inverse scramble, they start the solve already knowing part of the memorization.
Yes, that's precisely what happened at Euro 2010. I was judging an extra solve for some competitor (don't remember who it was). He started memorizing, but suddenly stopped, saying "Hey, I've already had that scramble !". So I went to the scrambling table to check the scramble: it didn't look like the original one at all. When I asked the scramblers what happened, they told me that since it was an extra solve, they simply had applied the inverse scramble. I had to explain them why it was a bad idea.