A fascinating recent post on the American Express OPEN forum argues that scientific research suggests that, for those seeking a eureka moment, allowing the mind to wander is better than trying to force continual focus. The post explains:
When you're awake, your mind wanders about 30 percent of the time, according to studies by psychologists from the University of North Carolina. Consciously you may be thinking about winning the lottery or scoring the winning run in your office softball league. But beneath the surface, your brain is often hard at work on big picture problems....So how can you encourage these golden flashes of dual function in the brain? Forget focus and allow your mind to wander. "Jogging, knitting or just doodling can relax the mind and set it off down a whimsical path," concludes the post.
While you're zoned out, the brain activates what neuroscientists have identified as a "default network". This area is especially active when people are reflecting on their personal experience or imagining the future, typical daydreaming preoccupations.
During complex reasoning, the mind switches to an "executive network", which is better suited to pursuing immediate goals. This top down system is more efficient at rational problem solving, but unlikely to produce any unexpected breakthroughs.
Occasionally both areas of the brain will be active at the same time. This state is critical to generating that eureka moment. John Kounios of Drexel University looked at images of the brain at the moments before someone realizes the answer to a puzzle. What he found was a flash of activity from both the default and executive networks, almost as if the two were working in concert to produce the inspiration.