Solving a cube, least memorization

Nomenclature:

I am not using the standard nomenclature here, because clockwise and counterclockwise  are counter intuitive when you're not looking at the face you're rotating.  I'm using symbols that reflect the rotation from the perspective of the person solving the cube.

I've also added line breaks to some sequences to emphasize patterns that make them easier to learn.

Symbol	Description	Movements
F	Front, the side facing you.	F↻ clockwise F↺ counterclockwise F2 halfway either direction
R	Right side	R↑ clockwise R↓ counterclockwise R2 halfway either direction
L	Left side	L↓ clockwise L↑ counterclockwise L2 halfway either direction
U	Upper, top side	U← clockwise U→ counter clockwise U2 halfway either direction
D	Down, bottom side	D→ clockwise D← counter clockwise D2 halfway either direction
B	Back, reverse side	B↺ clockwise  (It's reversed.) B↻ counterclockwise B2 halfway either direction

If you still prefer to think in terms of clockwise and counterclockwise, feel free to verify the movements as described above.

My drawings use numbers instead of colors because I want you to be able to choose which colors to work with first.  1 is the top color for the first 3 steps.  6 is the opposite side which becomes the top for the last 4 steps.  The other four colors swap numbers as needed.

Step 1:  Put each top edge in place to match the sides.

You should be able to do this without memorizing anything.  All the steps in this guide work better with practice, but in this case, it's practicing the techniques, not sequences.  Learn to identify which pieces go where and to find them.  Learn how the basic rotations affect pieces.  Once you have some top edges in place, try not to rotate the upper face very much.  Move pieces using the bottom and sides.

Step 2:  Put each top corner in place to match the edges.

Again, no memorization, just get to know the cube.

If you are struggling, this small sequence may help.  Rotate the bottom until the cube you want to place is directly below where it needs to be.

D→ F↻
D← F↺

If the square from the topspace is on the front side instead, do this:
D← R↓
D→ R↑

Both of these are part of the sequences from step 3, so you don't need to memorize them separately.

If the top corner is already on top, but in the wrong place, or rotated the wrong way, the best thing to do is to wait and put the others in place.  That will often fix the problem.  But if it doesn't, doing one of the sequences above will take the top corner to the bottom layer.

Finally, the top square you're placing my be on the very bottom.  Line it up under where it goes as above and do this to rotate it in place.

F↻ D2 F↺ D←  

Step 3:  Put vertical edges in place.

Find a missing vertical edge and the space it belongs in.  The opening that you're trying to fill should be front right, as shown below.

Rotate the bottom face so the vertical side of the out-of-place edge matches the center.  It should look like one of these two.  Use the given sequence.

D← R↓ D→ R↑ D→ F↻ D← F↺

D→ F↻ D← F↺ D← R↓ D→ R↑

If you find the edge piece is already in the middle layer, but backwards or in the wrong place, rotate the cube so it's in the gray space above and do either sequence.

Repeat that until all 4 vertical edges are in position.

Step 4: Last Layer - Rotate Edges

Now you're going to flip the cube over.  We want to make the squares marked 6 match.  Don't worry about the corners or positioning yet.  If you already have the 6's matching, skip to step 5.

Otherwise, rotate it until it looks like one of these.

This is the sequence for the left option.

F↻ U← R↑
      U→ R↓ F↺

For the other two options, you'll need to do the same sequence twice. Be sure to reorient the cube to match the first image the second time.

Step 5: Last Layer - Rotate Corners

This is the goal, to have all 4 corners rotated to make a solid face.

How close are you?  If you have all the 6's matching, go on to step 6.

If you have exactly one corner matching, orient the cube so it's in this position R↑ U← R↓ U← R↑ U2 R↓ This sequence may need to be repeated.

If you have zero or two corners matching, orient the cube so one of the mismatched cubes is in this position. You can use the same sequence, but you will need to do it more than once.

Step 6: Last Layer - Position Corners

Here is the goal, as seen looking at the last layer from above.  Pay attention to the 4 corner cubes marked in gray.  (The edges are in the last step)  If you can rotate them into position so they all match, go on to step 7.

Otherwise, there are two possibilities.  The simplest is has two adjacent corners swapped.  Position the cube so those two are top front.

R↓ F↻
R↓ B2
R↑ F↺
R↓ B2
R2 U→

If opposite corners are swapped instead, go ahead and use the same formula, then reorient yourself and use it again.

Last Step: Last Layer - Position Edges

Hey, if the cube is solved, you're done.  Show off.

Otherwise, if you're lucky, one of the 4 edges will match.  Put it on the back side as shown.

The following sequence rotates the three mismatched cubes clockwise.  a → c → b → a
If they needed to be rotated counter clockwise, you'll have to do it again.  OR, you can reverse the two U→ (clockwise) turns to U← (counter clockwise) and get it done in one sequence.

        F2 U→ L↓R↓
        F2         L↑R↑
U→ F2

If you haven't got any edges matching, just do the sequence anyway, and then you should have 1 matching edge to orient your cube for the second repeat.

Summary

With this simplified approach, there are just 6 sequences to memorize.

Step 3:

D← R↓ D→ R↑ D→ F↻ D← F↺

D→ F↻ D← F↺ D← R↓ D→ R↑

Step 4:

F↻ U← R↑
      U→ R↓ F↺

Step 5:
R↑ U←
R↓ U←
R↑ U2
R↓

Step 6:
R↓ F↻
R↓ B2
R↑ F↺
R↓ B2
R2 U→

Step 7:
        F2 U→ L↓R↓
        F2         L↑R↑
U→ F2

But you don't have to stop there.  There are many more sequences that you can learn to get faster.  You can already solve the 2x2 cube using only steps 2, 5, 6.

And, if you add 2 (and a reflection) sequences to this, you can solve the 5x5 professor's cube.  The 4x4 is more complicated due to "parity" problems, but you'll still need all the steps here.