shower 2
flickr/missnita
You're innocently shampooing your scalp when — bam! — an unforeseen solution pops into your head.
Psychologists have a name for that soapy eureka: incubation.
In the same way that an embryo incubates in an egg before becoming a full-fledged chick or a virus incubates in your body before becoming a full-blown disease, thoughts incubate in the back of your mind before popping into your awareness as fully formed ideas.
But this process doesn't seem to happen while we're actively concentrating on trying to find a solution. Instead, you need some element of distraction to properly incubate.  
One example, care of a Quartz report:
A study done by Dutch researchers Ap Dijksterhuis and Teun Meurs asked participants to invent new pasta names, after being prompted with five examples of fake names, all ending with an "i." The researchers found that participants who were given three minutes of a distractor task were much better at generating original pasta names that didn't end in "i" than those who were simply asked to sit and think of new pasta names for the same amount of time.
This finding — that casually occupying yourself as you putter around an idea increases the quality of ideas — has been confirmed in lots of other studies.
It's the same reason that "sleeping on it" helps with a major decision: Even if you're not actively weighing the costs and benefits of buying that car, renting that apartment, or taking that job, neuroimaging shows that your brain is still working on the problem while your attention is otherwise occupied.
With its trademark mix of warm water and familiar routine, showering provides an activity where you just barely have to pay attention to what you're doing — the perfect setting for incubation to bear fruit.