Some breakthroughs arrive after effort ends because the mind finally has room to connect things.

Concentration is good at pursuing a line and poor at noticing connections between distant things. That second kind of thinking happens when the pressure lifts, in the shower, on a walk, in the days after you stop pushing. So the insight arrives after the effort not because the effort was wasted, but because the material needed a quieter room to arrange itself in.

When you are stuck, put the problem down properly rather than half-down. Do something unrelated and physical. The pause is part of the method rather than an admission of failure.