# What does it mean for a Rubik's cube to be perfectly scrambled, and how do you reach it? Without doing a checker board

Can you picture the perfectly scrambled cube? A Rubik's Cube that's perfectly scrambled? How would it look? Is it even possible?

The quest for the Perfect Scramble is now upon us, and I challenge you puzzlers to find the state that is "perfectly scrambled" and how to do this scramble, starting with a solved cube, in minimum moves.

But first, before you can even begin to figure it out, mathematically prove it, or even brute force it, you must define it. What exactly is a perfectly scrambled cube? there's 6 faces to the cube, and 9 squares on each one, so how do you define the perfect scramble?

The perfect scramble is the scramble farthest away from the perfect cube. What would that look like?

The quest for the perfect scramble is upon you. It's up to you to solve this Rubik's crisis! Can you do it?

The "most scrambled cube" is any configuration that requires the greatest number of rotations to solve the cube using a perfect algorithm.

Although unknown, this algorithm is hypothetically called God's algorithm and in fact the maximum number of rotations, called God's number has been found to be 20.

The list of such 20-move cubes is given here.

To define the "most scrambled-looking" cube, consider any legal cube configuration where:

1. each face displays all six colours
2. for each face: upon blacking out the cells of any $n$ colours, $n \leq 4$, the resulting $3\times 3$ grid must exhibit no horizontal, vertical, diagonal, or anti-diagonal symmetry
3. no line of 3 identically-coloured cells may appear horizonally, vertically, diagonally, or anti-diagonally on any face

Alternatively, we might gauge scrambledness via empirical observation as follows:

1. Given the population, $H$, of all capable human beings on Earth, let $S$ be a subject drawn from $H$, and conduct the following experiment $\forall\; S \in H$:

1. Explain the nature and objective of the Rubix cube to subject $S$, and provide a "play" cube for familiarization purposes.

2. After a minimum of 5 minutes of continuous play, present $S$ with a complete set of 43,252,003,274,489,856,000 rubix cubes, one per each legal state of the contraption. For every unique subset of 3 cubes, have $S$ rank the cubes in order of increasing scrambledness per his/her subjective preference.

After each evaluation, tally $0$ for the cube ranked "least scrambled", $2$ for the cube ranked "most scrambled", and $1$ for the remaining cube.

2. After all subjects in $H$ have completed all tallies for all subsets of all possible cube states, assign to each cube state a datum $s$ that is the sum of all numbers tallied on the state. $s$ will be a non-negative integer $<$ 1.7$\times$1069.

3. Let $s_{\rm max}$ be the $s$ datum of greatest magnitude computed in step 3. Let $C$ be the set of all Rubix cube states for which $s = s_{\rm max}$. We conclude that any cube $c \in C$ is "maximally visually scrambled" with 95% certainty.

• No, this is the opposite. We can uniquely mathematically define the set of cubes that are furthest from solving. If you want "furthest-looking", I suppose an obvious answer would be a cube where every face contains all six colours, and no more than two of any colour, and no more than two colour pairs in common between any two faces. But even that's pretty subjective. :-\ – COTO Nov 23 '14 at 16:37
• @warspyking The problem is: how do you want to define "futhest-looking"? 20 is the furthest. – d'alar'cop Nov 23 '14 at 16:45
• @warspyking, I can think of at least two or three ways to define "visual scrambledness" of a cube, but they're all algorithmic and impossible to compute in a reasonable amount of time. I can append them to the answer if you want. – COTO Nov 23 '14 at 16:52
• @JulianRosen: It's a statistician's joke. If you actually carried out the absurd process of surveying every single human being on Earth, your sample would equal your population, and margin of error would necessarily be zero. Hence the irony of going to such ridiculous extremes but claiming an error bound no tighter than 5%. – COTO Nov 23 '14 at 19:32
• @COTO For what it's worth, I almost fell out of my chair laughing when I read it – Julian Rosen Nov 23 '14 at 19:35

There are (at least) two ways to measure how far a cube is from being solved: the quarter-turn metric (which counts the minimum number of quarter turns of faces needed to solve the cube), and the half-turn metric (which counts the minimum number of face turns needed, be they quarter or half turns).

In the quarter-turn metric, a cube can be at most 26 steps from being solved. This goes down to 20 steps for the half-turn metric. Surprisingly, only a single position is known requiring 26 quarter-turns to solve. Hundreds of millions of known positions require 20 half or quarter turns.

Images of cubes requiring the maximum number of moves, as well as descriptions of how to reach the positions, can be found here (in the quarter-turn metric) and here (in the half-turn metric)

Here's another perspective. Realistically, humans have just as much trouble with a position which requires 18 half turns as they do with a position which requires 20 half turns. Therefore I do not see why exactly the actual distance a position is away from the solved state should be considered "perfectly scrambled" (the first two responses mentioned this), as when I hear "perfectly scrambled", I think of the difficulty for a human to solve it...it's not like humans have a method to solve a cube optimally every time (if they can by chance solve one optimally in a fewest moves competition).

Therefore, I define a perfectly scrambled cube state to be any state which doesn't allow any extra pieces to be solved than intended during the solving process. It's a scramble which we call "unlucky" or "full-step". Since there are different human methods for solving a cube, in reality, a perfectly scrambled cube for one person will be an easier one for another person, and therefore a perfectly scrambled cube is unknowable and yet occurs frequently for everyone.

The only way to determine if a scramble is perfectly scrambled for YOU is for YOU to solve it with whichever method you use. Even then, it's questionable whether a scramble is "unlucky" for you should you have chosen to solve the pieces in one step in a different order, for example.

Therefore the conditions for a perfectly scrambled cube occurring is:

1. Unique to the individual,
2. Dependent on the order in which that individual chooses to solve pieces in each stage of the solving process he or she uses.

I therefore can understand why the first two responses were mentioning God's number, as that is the only answer which can solidly answer this question, but I do not believe the two are related. Therefore I have answered an inexact question with its corresponding inexact answer.