Lots of problems can be solved with a good night’s sleep. It’s hardwired into us. During sleep, the brain takes the day’s events and decides what to store in memory. This is an evolutionary advantage because our ancestors could remember where the food was found.
If you had been working on a problem, the brain assumes that it is important to your survival, so it makes attempts to solve that problem and store the solution in memory. There was a recent episode on PBS’ Nova about this process. It monitored the brains of rats and showed how they tried to solve a previously unknown maze to get to food. Those that had slept after exposure to the maze performed much better than those who just kept trying without sleep.
One of the other findings was that the very young and seniors should be taking short afternoon naps in order to increase their retention and to not worry about problems so much.
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