I can recall one very specific instance when a frustrating problem was solved miraculously during sleep. I found the answer when I awoke one morning, but when I asked at the office for materials to check it out my management objected.

When I told them that I found the plan beside my bed when I woke up, and "It must have come from God, telling me what to do," they apparently believed me¹, even though the "answer" was written in ballpoint pen on a Kleenex rather being chiseled into the standard stone. They agreed to let me have the $3.72 worth of parts for the test.

It worked perfectly, but was never used because the supplier who needed to build it said it was "too complex" for their people to build. (Their real objection was the $3 worth of parts, since they'd submitted a fixed-price bid.)

The actual answer was that while asleep I'd grabbed the pen and drew the circuit diagram - without waking enough to clearly remember doing it in the morning. (Although I did wake up with a clear feeling that a great burden had been lifted.)

¹ This was in Kansas, so it was a common enough explanation when that particular management didn't quite understand something "technical."

John