During my time at university studying pure mathematics I had an interesting experience of doing a challenging sheet of combinatorics problems during a vacation. Every day I attempted one question and got stuck on it. Then the next morning I woke up knowing the solution. It was a recurring thing: this happened every day for about 2 weeks until I had solved all the problems.

For me this a big eye opener about the importance of sleep and relaxed thinking to solve challenging problems.

In my 40's I could go to bed with a complex software design or implementation problem I was wrestling with. Consciously word a cogent and succinct question that I needed answered, sleep on it, and then in the morning, I would be still and mentally ask, "well?" Not meditating or anything, just be quiet then and listen.

And, in very deadpan style, after a few seconds (as if to choose one's words carefully), some answer would come to me audibly in my voice in my mind.

"Have you tried X?" No, I hadn't tried X, and holy smokes that was a workable approach! Sometimes, it would tell me to go back to some bit of code or configuration I had moved on from and tell me to go back and focus on that, it was almost always right that there was where I had goofed up. I experimented with posing multiple questions and follow up questions. I even asked it how it was that these answers were derived.

Strange to reread the above and refer to my own thoughts as 'it'. They were bidden ideas that came from me for sure. But, I disassociate from them because I have no memory of the chain of thought that led to the responses.

There's a lot going on upstairs, higher mind stuff. I am older now, and I no longer experience this phenomena. Have I lost it to age, or have I integrated it somehow into my conscious mind?

I often wake up realizing I wrote a handful of bugs the day before. They’re obvious to me in the morning even before I sit down and open my laptop.

Sleep is a strange and magical thing.

In French, there's a saying: "la nuit porte conseil". Roughly translates to "the night advises", and it means it might be better to sleep on it.

I recall my father (also a mathematician, incidentally) often repeating this to me.

Yeah, when you are stuck, put away that red bull and step away from the keyboard, kids.

People are so different. When I was in college, if I had an unsolved problem, I could not fall asleep.

That's awesome! I had a somewhat similar experience (shared previously [0]):

> I proved a topology theorem in a dream once.

> Before I went to sleep, my inability to prove it had been bugging me all day long, and I suspected it'd be featured on the next morning's (way too early) final exam for my university course. I solved it in my dream, woke up, wrote on my whiteboard what I remembered and sure enough, it was correct. I worked it a few more times to cram it into my memory before running to my exam.

> To my great delight, the ability to prove that theorem was featured heavily in one of the exam's questions, and helped me do quite well on the exam overall.

[0] https://news.ycombinator.com/item?id=40651913

Interestingly, I observed the same when I was practicing the drums. I would fail multiple times to reproduce a drumming part, sleep on it, and succeed on the first try the day after.

Can confirm this level of problem solving.

Had physics problems to solve and can remember to this day when I woke up in the library after I got exhausted from not solving the last one, that my subconscious discovered during sleep that I missed that certain vectors were orthogonal (which was the necessary key insight to solve it).

I can confirm - I woke up to the resolution to my two hardest problems during PhD. Three, if you count "I should look for this kind of inequality" (which did turn out to exist), but I think that's more of an _idea_ than a solution.

The hard part is paying attention to it. With enough attention your mind will fix it.

AFAIK this is called sleep consolidation.

This is exactly how I learned programming.

10-hour days practicing. Full night sleep afterwards.

"Sleep on it?"