Have you ever had a "middle-out epiphany"?

The usual way it works for me is packing in enough information about the problem, the context, and relevant background/theory as I possibly can - to the point of obsession. Then once I've established a constant background hum of thought, I have to encourage the epiphany to arrive.

Monotonous physical movement is an epiphany trigger. Activities like running, washing dishes, and showering all set the stage for the epiphany to arrive. It can take a week or two when it comes. If it doesn't then I take a break for a month and try again.

For various reasons, I was preoccupied with three different thoughts in 2005: the Euler equation (relating vertices, edges and faces), transforming planes with 4x4 matrices, and trying to optimize ray-triangle intersection. On the way to a company meeting, I was sitting on a bus, looking out the window, watching the scenery go by while turning these three topics around in my head. These pieces (which I didn't think were really related) suddenly started fitting together, and I came up with a radically new way of testing ray intersection against an object defined by a mesh of triangles. It ended up being successful beyond my biggest hopes. Eventually got patented by Microsoft, but the patent's now expired. Still my proudest discovery.

So you mean a Eureka moment? https://en.wikipedia.org/wiki/Eureka_effect

While a lot of progress is made by chipping away at something, sometimes an idea arrives apparently (very nearly) all at once. At least one of the ideas in patent GB2594749, now in > 10E5 devices, came about arguing with a colleague about why something simply wasn't possible...

While helping somebody with an assigment during uni studies I realized binary counting is equivalent to computing all the permutations. We used that fact to solve one of many problems in an allnighter.

Not very impressive but it felt deeper to rediscover by oneself.

For me this is common. Not so much in the "saved the company" sense, but rather in the "oh, I know how to solve that" sense.

And only occasionally for big things. More often for little things. Bug fixing. Feature design and so on.

I seems when I stop looking at the problem so hard, the solution works itself out subconsciously, and is waiting for me the next time I think in that direction. Usually in the shower :)

Of course this is just the beginning of the actual solution. The work is in taking the idea and then turning it into something useful. Ideas by themselves are valueless.

Yes. I'll check back here in a few months when it ships to everyone on the internet (if it all goes well) :)