He'd been asked by the king to find out whether his crown was pure gold without destroying it. Getting ready to step into his bath, as Archimedes put his foot in the water, he noticed how the water rose as his foot went in. That's when insight struck. He saw that all he had to do was submerge the crown in water to determine its density.
He got so excited that he shouted "Eureka!" – which means "I have found it!" --and ran out into the street, forgetting he wasn't wearing his toga. Well, ever since then, people have noticed these special moments when insight suddenly strikes. The challenge lies in knowing why these moments happen and what can be done to encourage them.
From scientific research on the brain, it seems that our usual logical thinking approach is handled by the left side of the brain. But those sudden insights appear to come from the right side of the brain. And when those aha moments come, your brain does literally "light up" – there is a burst of high-frequency brain waves. It seems that the right temporal lobe of the brain is used for drawing together distantly related information, so when it all comes together, it's like all the pieces of a puzzle suddenly falling into place, and you've got your "light bulb" moment.
So how can you encourage these sparks of insight? The most important thing seems to be not to think about it! Using the left-brain logic sometimes just doesn't help us solve problems. You can literally "think" too much. Instead, try doing something else, put your attention elsewhere. Take a walk, talk to a friend, do a chore. Maybe even take a bath, like Archimedes. Just be sure to put on your "toga" when insight strikes!