What's the most efficient method to scramble Rubik's Cube to consider it as sufficiently scrambled?

Question

I am enthusiast of rubik's cube but recently I got a question and I thought more opinions would help to settle this discussion in my head. So I scramble the cube, just random twists multiple times and then I just solve it. BUT how to know if I scrambled it "enough" to consider it as challenging. Is there any sort of "levels" of difficulty for scrambling or correct way to do it so there wouldn't be a question that it's scrambled enough. The reason I ask is whenever I use rubik's cube solution guide for sake of curiosity I almost every time can skip all the beginner's steps and start from the middle of guide. That's what got me questioning if I scramble enough. Any information is welcome!

Answer 1

The scrambling methods that gabbo1092 describes in his answer are great and very practical, so I’ll delve into the theoretical side a bit more.

It has been proven that at most 20 moves are needed to solve the Rubik’s cube, and the majority need 17 or 18 moves. That does not mean however that doing 20 random moves on a solved cube is enough for a good scramble. Sure, every possible position can be reached, but not all of them will be equally likely. Positions at distance 20 are not as likely as there are fewer ways to reach them than positions a bit closer to the start position. Unfortunately I don’t really know how many random moves it takes for a relatively even distribution.

For this reason the official competitions use a scrambler that chooses a random position (uniformly, so that every position is equally likely) and then generates a scramble for that particular position by solving it and taking the inverse of that solution.

That’s all very well for the standard Cube, but for other puzzles no such solvers exist, so how do you determine how many random moves is enough?

The very minimum that you would like to have with a scramble is that every pair of adjacent pieces was separated at least once during the scramble. There is a simple way to estimate how long that takes. Imagine you have pieces of sellotape holding together every adjacent pair of pieces. For the 3x3x3 you’d have 48 bits of tape, 8 for each of the six outer layers. Every time you want to do a move, you’ll need to use a knife to cut the tape first. The more moves you do, the fewer bits of intact tape are left, until eventually they have all been cut through. If you assume that throughout the scramble the remaining bits of tape are fairly evenly distributed, then each move reduces the number of bits if tape by $8/48=1/6$, leaving $5/6$th intact. After $n$ moves the $48$ bits have been reduced to approximately $48(\frac{5}{6})^n$. This becomes less than $1$ when $n$ is $22$ or more, so we need at least $22$ random moves for a good scramble on a 3x3x3 cube.

For larger puzzles this method gives a reasonable lower bound for the scramble length, though you may well want to do a few extra moves just in case. Unfortunately humans are bad at randomness, so if you don’t use an external source for it (e.g. dice, computer) then you probably need to do many more for a good scramble.

Answer 2

Not sure if this is what you had in mind but a method to have unique and "sufficiently scrambled" cubes is to use a computer generated scramble similar to ones used in competitions. If you search online you should be able to find at least one that works well for you.

For competitions there are specific rules for what qualifies as sufficiently scrambled that there random sequences are forced to consider. Those rules can be found here.

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Simple Alternative Methods:

These methods are for if you struggle to follow the computer generated sequences or simply want a faster way to scramble so you can get back to solving. They won't always be the most scrambled or random, but I've found they work well.

Have someone else mix it up
In my experience, it tends to give a more random mix if the person doesn't know any steps in solving because they don't know what to look for so they can't accidentally set up steps.

Mix it up with your eyes closed
Mix it up for a while and make sure to move how you hold the cube to disorient yourself in a way. This should give a relatively random mix up since you can't tell what's happening. This also works well if your practicing timing and start your inspection or solve time as soon as you open your eyes.

Start solve from a different side
This can work if you always start on the same side after mixing. So if you want to mix things up a bit more or if the side you normally start with seems to be mostly solved to begin with, purposefully choose to start on a different side that seems less solved to practice those first steps.