Yes, it's certainly possible, but not very easy when you've never done it before. First of all you need to have a good understanding of the puzzle itself and what everything does. What I mean by this:
When you rotate a layer 90 degrees, it does the following:
- 4-cycles four corners without changing orientation
- 4-cycles four edges without changing orientation
- rotates the center 90 degrees (which is unimportant for a regular Rubik's Cube which lacks center orientation, but is important for so-called super 3x3x3 sticker mods/shapemods)
This might look trivial, but it's important to know when solving it and to know how pieces move back and forth.
I personally haven't solved the 3x3x3 Cube on my own. But, I am a Twisty Puzzle collector and have solved other puzzles. I'm not an expert in solving if I'm completely honest, but my general approach for solving a puzzle is as follows:
- Do some random sequences until I find a sub-algorithm that does something useful. (Useful can be anything, like a double two-swap, a five-cycle, etc.)
- Then I use setup moves before or during that sequence to create other (more) useful things (usually a three-cycle or orienting some pieces without moving anything else).
These sequences (and the setup moves before / during that sequence) are combined in commutators. Commutators are generally as follows:
A B A' B'. With what I've explained above,
A would be the 'useful sequence',
B the setup move,
A' the reversed of that sequence, and
B' undoing the setup moves.
For some concrete examples of some puzzles I've solved myself and how I came up with the algorithms I refer to this reply I made on the Cubers-reddit a few weeks back.
One of the puzzles I mention in the post I linked above is the Geared Mixup:
(Click for a larger image)
At one part of the solve I had to orient the gears. When I tried the following
[R2 U2] (3x) I noticed it would orient every single edge-gear on the puzzle 180 degrees. So, I used some simple setup moves in between to form the following algorithm:
U' // 1
[R2 U2] (3x) // 2
L // 3
[R2 U2] (3x) // 4
L' U // 5
U' as setup move puts four centers at edge-positions and vice-versa.
- When I now execute the sequence I found, these centers in edge-positions will be rotated 180 degrees instead of the edges that were in their current positions. Since centers lack orientation this doesn't matter, and the edges are safely stored at the center-positions.
- Now I do an additional rotation with the same goal as the first
U'. The reason I do two is so with the full algorithm will only rotate two gears by 180 degrees. Would I have just used the
U' without this
L then it would rotate four gears instead of two, which is a lot more situational and harder to setup.
- Now I execute the sequence again to re-orient all the edges by 180 degrees again. Because two of the edges are still stored at center positions, they aren't oriented back.
- Now I undo the setup moves to put the centers and edges back at their original positions.
The total result of the entire algorithm is that I've oriented the gears at the top back and bottom left both 180 degrees, without orienting or moving anything else (except for orienting some centers, but those lack orientation, so you don't notice this).
(It's a more advanced commutator in the form of
B A C A' C' B'.)
You can also combine commutators with other commutators. This is also usually something I do. For example, usually I'm able to find a double two-swap to solve almost everything, but then end up with three pieces left to three-cycle. In that case I try to find a sequence in the same form of
A B A' B' with the double-two swap and setups to create a three-cycle needed. The total algorithms usually end up very big (like 50+ moves), but it's still satisfying to be able to solve a puzzle completely on your own.
Sorry for the wall of text. If you've never solved a puzzle yourself it might be pretty difficult, but I know a lot of people that have been able to do it. My suggestion is to learn about commutators, then get your Rubik's Cube and a pen and pencil, and write everything useful down.
Also pick a general solving approach (which could be decided later and based on the algorithms you find). There are a lot of different methods for solving a 3x3x3 out there. Usually it's done layer-by-layer, which is probably also what you've used with the tutorial, but other approaches could be edges-first/corners-last; corners-first/edges-last; making a 2x2x2 block first; making two 1x2x3 blocks first and then solve the middle and top layer; etc. Loads of different approaches and algorithms to find.
For now I wouldn't aim for efficiency, but just for accomplishment for being able to solve it on your own.
Good luck. :)