The brain has different strategies for problem-solving, and “daydreaming” is one of them. Brain scans show that in mind-wandering states the brain is actually more active, and is engaging more divers brain sectors, than when it is doing an analytical task.
Analytical reasoning is primarily a left-hemisphere activity. In tests that involve solving puzzles, some people approach the task analytically, while others are more likely to use an “insight” strategy, which brings the right hemisphere into play. This hemisphere is instrumental in making associations between the divers elements of a problem; in other words, it sees connections. The operation of the right hemisphere often happens at a sub-conscious level, while the mind seems to be wandering.
Insights into problems—sudden, out-of-the-blue realizations—appear on an EEG as a burst of gamma waves emanating from the right hemisphere about ⅓ of a second before the person consciously experiences the insight. Researchers could tell which subjects were likely to solve a problem using the “insight” strategy by a certain pattern of neural activity in the right frontal cortex.
The person is unaware of all this background activity before the insight arrives. It happens when the mind seems to be disengaged from the problem because the left hemisphere has abandoned trying to analyze it.
This doesn’t mean that the best way to deal with problems is to ignore them. It does imply, however, that it can be useful to periodically disengage from consciously thinking about a problem and give the mind free rein. It will continue to look for connections in the broader mental space that’s freed up when the intellect is at rest.