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I've read anecdotes about people who solve technical problems in their dreams and the concept of getting a few more mental hours out of the day is very appealing.

Have you never experienced it? I've found it's very common that I'll sleep on a problem and have solutions in the morning.

Yep, I solved several Math and Physics problems for school (I used to get into competitions), understood a martials arts technique (it was taught to us but nobody could make it work. In my dream I fought someone but I saw everything upside down, and there I figured the technique was very similar to other one we already knew, but upside down), and solved many programming problems I had when I started learning that.

I also feel like my naps defragment my brain even when I'm not 100% unconscious. I go through things that happened that morning or the day before and kind of classify everything. Sometimes I understand how the people I interacted with felt. Things that in the moment they occurred I couldn't appreciate. Like when you consciously try to think things from the other person's perspective, but dreaming sometimes forces me to do that.

I've experienced it occasionally. More often, I've (at least had the impression that I've) solved it in the night, been aware of solving it, and then forgotten the solution by morning.

There is a short window between sleeping & waking where recall is possible, you need to capture the thought immediately. There's a book on this topic, http://www.nytimes.com/2010/07/27/science/27dream.html

"Using dream incubation for problem solving, Dr. Barrett, the author of “The Committee of Sleep,” which expanded on her initial research, asks patients to write down a problem as a brief phrase or sentence and place the note next to the bed. Then she tells them to review the problem for a few minutes before going to bed, and once in bed, visualize the problem as a concrete image, if possible.

As they are drifting off to sleep, the patients should tell themselves they want to dream about the problem and ideally keep a pen and paper, and perhaps a flashlight or a pen with a lit tip, on the night table. No matter what time they wake up, they should lie quietly before getting out of bed, note whether there is “any trace of a recalled dream and invite more of the dream to return if possible.” They should write down everything they remember. "

http://en.wikipedia.org/wiki/Hypnopompic, "A hypnopompic state is the state of consciousness leading out of sleep ... When the awakening occurs out of rapid eye movement (REM) sleep, in which most dreams occur, the hypnopompic state is sometimes accompanied by lingering vivid imagery. Some of the creative insights attributed to dreams actually happen in this moment of awakening from REM"

With segmented sleep, there are two daily instances of sleep-wake transition, so the chance of capturing notes is doubled, http://www.polyphasicsociety.com/polyphasic-sleep/overviews/...

I've had similar experiences with the opposite result. A number of times I've believed I've solved a problem with a eureka moment in a dream only to wake up and realize my dream-solution was complete nonsense.

I've solved several problems, written several symphonies, and even had a blinding epiphany or two about how to solve all the world's problems that I've managed to drag up into the waking world. All were either complete nonsense or useless. (e.g., the solution to my graph theory problem is not to force feed my homework to a goat and then look for the answer in the pattern of the resulting droppings). I think it's news when someone finds a good solution in their dream precisely because it doesn't happen often.

Sleeping on the problem and having fresh insights of course is such a common experience as to be pedestrian.

When I was a kid I was stuck on a puzzle in "Ecco the Dolphin" (for the Sega Game Gear). I woke up suddenly in the middle of the night after I got through the puzzle while dreaming. Beat the whole game by morning.

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