A valuable method of thought is one that Z. A. Melzak has called "bypasses"- also known as "similarity" transformations in a mathematical context. In brief, when facing a problem we cannot solve, we look for a way to map it into something we can handle, solve that task, and then map the solution back into the original domain. Melzak's archetypal example is for the challenge that a wall presents- we can tunnel down, go across, and then tunnel back up to get to the other side, if we cannot go through the wall directly. In the terms of an equation, we can convert the hard problem H into the easier problem by a map; E = MH; we solve E and then undo the map M with its inverse, 1/M. The result is the transformation M * H * (1/M).
This method of bypasses appears constantly in problem solving. To reorder a scrambled Rubik's cube is tricky, because the moves that we can make all do complicated things and tend to mess up the parts of the cube that we have already gotten into place. But an approach based on the bypass method can help. We make a transformation that scrambles the cube but leaves one slice of it mostly unchanged- call this "M". We then do a simple change to that slice, and then undo (invert) the original transformation (performing 1/M). We can thereby control the complexity of the whole operation so that it does something manageable- such as swap two cubelets, or flip a pair of faces of edge cubelets, or twist two corners in opposite directions.
Similarly, bypasses can apply to many ordinary problems. We have to work with a difficult person to get a job done- perhaps we can instead get an intermediary to translate between us, someone who is comfortable on both sides; or perhaps we can divide the task so that each of us only has to do a minimal amount together.
Wednesday, April 14, 1999 at 06:22:39 (EDT) = Datetag19990414