I recall back in the early 80s when the cube first came out, I spent a sad Christmas solving the cube. My technique (after having completed the first layer), was to map out the relative position of the colours – going through a set move sequence ‘breaking’ the top layer (then getting the top layer back), and observing the change in position of the colours. Then knowing how the colours would move in relation to one another (following the set sequence), I positioned the requisite colours at the required location prior to making the moves – and then put the cube through the set move sequence.
Having completed the top two layers, I then used a similar process in first correctly positioning the four base corners, and then the four base sides to compete the cube. In all I required four sets of move sequences to solve the cube. I should point out that the longest of these move sequences was over 15 – and it has been calculated that the cube cannot me more than 17 moves from complete.
Using my technique, the fastest I ever managed to complete the cube was around 3 minutes.