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Alice needs a strong password, but she has terrible memory - so she have decided to hide it in a Rubik's cube she happens to carry around all the time

  • Alice starts with a solved 5x5x5 Rubiks Cube
  • She does exactly 7 moves (rotation of one slice by 90/180/270 degrees)
  • After those moves she reads the password as starting letters of color names on the side with white center, read row by row from top to bottom, where the side with red center is in the front side.

It will look like "wwrgrbbwwwbowowooyyywyoor"

Can you get Alice password just by knowing the side with red center?

ryrwrryrgrryrwrgbbrbbbbbb

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    $\begingroup$ I take it that the image has nothing to do with the string given?? $\endgroup$
    – NL628
    Mar 22, 2018 at 1:14
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    $\begingroup$ the given string is just an example, unrelated to the image. The image has its associated string as 'alt' text $\endgroup$
    – Hurda
    Mar 22, 2018 at 7:15
  • $\begingroup$ Can you confirm the colour scheme? Is it U=white F=red, R=blue, D=yellow, B=orange, L=green ? And is the red face we are given in that orientation, i.e. with the white-centred face adjacent to the top edge of the image? $\endgroup$
    – Jaap Scherphuis
    Mar 22, 2018 at 8:51
  • $\begingroup$ It is a standard rubik cube, with no info given about final orientation - it should be obvious from the moves $\endgroup$
    – Hurda
    Mar 22, 2018 at 10:37
  • $\begingroup$ There are several single moves which affect only white side but not red side. There are also several series of moves which leave red completely unaffected, but yield different white sides. Seeing the red side will absolutely reduce the entropy of the white side, but there is no distinct password. For instance, just rotate the slice with orange middle. Maybe I am missing something? $\endgroup$
    – Carl Löndahl
    Mar 22, 2018 at 12:35

2 Answers 2

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Using the description in the question, I wasn't able to decide which way is "up" on the white side, so I would have to take four guesses, but I think the cube looks like this:

enter image description here

This is probably unique, my Most Rigorous proof is that it was pretty damn difficult to get there in 7 moves :-)

Here are the 7 moves: (with the cube upside down, because that's how the simulator I used places the initial cube.)

enter image description here

The third and the fourth move can be done in a different order, of course, but that still produces an identical cube. Any other changes to the sequence would not produce the required red side. The only other variations I found that get anywhere are

  1. Constructing the pattern the other side up
  2. Omitting the second move
  3. Both of the above

Sadly, it is only possible to get a (partially) mirrored version of the required pattern in any of those ways. (Proof: spent way too much time trying to make it work.)

Therefore, if there is another way to construct the red face in 7 moves, it has to be with some completely different moves altogether.

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    $\begingroup$ Best. Proof. Ever. $\endgroup$
    – Sid
    Apr 6, 2018 at 11:16
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    $\begingroup$ very proof. much rigorous. $\endgroup$
    – Rubio ♦
    Apr 6, 2018 at 11:17
  • $\begingroup$ How did you make the gif? That's so cool! :P $\endgroup$
    – Mr Pie
    Sep 9, 2018 at 8:08
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It is better if Alice stores her password on a 3x3, as we can

Guess of the centers, as they are fixed, and find out the other pieces, by tracing the 7 turns, and just having one face as the input after that.

In the case of a 5x5, it gets tricky, as

The number of possible slice moves increases, and also the the centers of two kinds (x centers at the corner fringe, and + centers at the edge fringe), cannot be deduced, even if you are given the information of 5 sides, and one side is hidden.

The reason being

That there are nifty amount of 2-cycles that can be put up on this centers, making them untraceable unless we have the visual of all the 6 sides.5x5 center information

On the other hand

A 3x3 3x3 cube, has 4x10^17 possibilies, letting us store even a 10 digit ASCII code password quite easily, if we form a map between each state, and a string of ASCII characters (this will be a tedious task), but a 3x3, or multiple 3x3, solves the problem.

If your password is quite long,

Then there is one more fantastic solution, you can have 2 cube states added onto a single 3x3 cube : eg, U D M B' U' S R' S' R U B M' U' D' + F E F2 R' S' R S R' E' R F, and you can retrieve it by some computational factoring algorithm, the same way we try to factorise big prime numbers during encryption.

See this video for more clarity on what I mean by adding cube states: Adding cube states

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