When we convince ourselves we've already solved the problem before even attempting it, our brains just aren't as focused on the task.
Excellent point, and I'm glad you bought that research up. You beat me to it! The scenario is the same as the kid who focuses completely and thoroughly on getting a present, and can think of nothing else. Upon buying him the present, after a day of using it, the kid almost completely loses interest and ceases to think about it much at all. This isn't because the toy is less interesting then he thought, but because the knowledge of having it took away his focus on it.
A problem arises when a person has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action, then there has to be recourse to thinking.
In studies, people who were mostly students of universities or of colleges, were given various thinking problems, with the request that they think aloud. The subject who is thinking aloud remains immediately directed to the problem, so to speak allowing his activity to become verbal. This is a technique I frequently use to problem solve. It's a form of "instructing" myself, but also challenging my thinking and being my own "devil's advocate.
What has also been born out by research is that the classic thinking of solving puzzles increasing your cognitive abilities and staving off dementia has also been shown to not be quite right. They found that what did was, in effect "train" the subjects on how to solve that "particular" puzzle faster and more effectively, but did little for overall cognitive development, which is a global brain process that can assist in broad problem solving. Interestingly, what the research did find is the single best thing that was shown to help with this was taking long walks or runs. This makes sense, as when we do this, our brains are immediately engaged in all our senses as we analyze our environment, seek out the interrelationship of what's around us, remember where we are, and plan where we're going. Having evolved from a largely nomadic people, this makes evolutionary sense.
In short, if you run into problems with your task, be in puzzle building or puzzle solving, or even other dilemma's with work, family or finance, then put the puzzle on the shelf, go outside, and go for a run/walk for at least 30 to 45 minutes. You will potentially find moments of inspiration to see you through or, at least, reset your mind so that you can redirect yourself to the task.
This issue is of great interest in artificial intelligence (AI), as if we can learn the techniques the brain goes through to problem solve, then we can design machines that can do that to! It is obvious that the memorization of a great number of facts will not take you all the way.
Here are some techniques that can help:
Abstraction: solving the problem in a model of the system before applying it to the real system (ok, so if I ignore the shapeshifting of this mod and solve it just like a cube, then it will work!)
Analogy: using a solution that solved an analogous problem (ok, so this corner turning octahedron looks just like a 3x3 mod, so I'll use the same algorithms as a 3x3 to solve it!)
Brainstorming: Suggesting a large number of solutions or ideas and combining and developing them until an optimum is found (ok, so if I want a Transformer cube, I'll just post the problem on tp forum and and see what they come up with! Hint hint!)
Divide and conquer: breaking down a large, complex problem into smaller, solvable problems (I'll just reduce my 4x4 to a 3x3 and then solve it as such!)
Hypothesis testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption.
This last one is in essence what Tony was talking about, which is a viable problem solving technique.
Sorry for the long winded response, but this is a field I'm very excited about!