Consider the following problem:
Four distinct lines A, B, C and D are given in the plane. A and B are parallel, and C and D are parallel. Find the locus of the point moving so that the sum of its perpendicular distances from the four lines is constant.
You might like to try it before reading on.
What determines whether we become 'involved' in solving a problem, or not? First, and possibly foremost for students, is whether one 'has' to solve the problem - whether at the behest of the teacher or an examiner.
Suppose then we have decided to try the problem. How does one then become 'involved in it'? For me, there is a critical period when I am not sure whether the problem is of genuine interest, or not. At this stage, I am not 'invested' yet - that is, I have not given myself to the problem - I remain relatively neutral.
I am ascertaining whether the problem is at the 'right' level of challenge. This is perhaps dependent on whether there are any approaches, insights and conjectures available, alongside something intriguing. There may be a tension that is demanding to be resolved. All of these factors have the effect of mobilising the energy required to get involved in the problem. If they are not present, I might decide not to continue working on the problem, or to give up.
Should the energy be available to continue working on the problem, it becomes objectified through trying examples, sketching diagrams, making conjectures. When choosing what to explore, I pick examples that contains some of the complexity of the problem, but not too much, with the option to increase or reduce the complexity and/or generality of examples as the need arises. Often it might be illuminative to explore a special case, if there is one.
Your sketch here might have included two pairs of parallel lines of different 'widths' and orientation, forming a parallelogram. Or perhaps you drew the more special case of a rectangle or square? Did you start by exploring the movement of the point that is equidistant from all four lines?
We gradually become more involved in the problem. Releasing vague thoughts/ideas/conjectures from the mind into the world (on paper, in words) may make it easier to regard to whole problem, thus mobilising further energy.
As we get more involved, it becomes more difficult to maintain the initial neutral state. We may get attached to particular ideas, enabling and disabling emotions might surface.
Should we become too attached, it may be beneficial to return to the contemplative state, the essential problem solving state. It provides a lightness through which we are able to maintain the whole in its complexity without grasping, thus improving our chances of receiving insight.
Hopefully at some point insight appears. You may believe that you have solved the problem. But how do you know? Have you reached the full solution? [Spoiler alert below!]
You might have conjectured that the solution to this problem is the area between the parallel lines... but is this the whole story?