[furyofantares]

That was exactly my experience and the reason I asked the question :)

The answer just appeared with zero awareness of any effort or reasoning that may have lead to it -- the only reasoning I perceived was in verifying the solution. So I got curious as to how intuition could solve this problem roughly on its own, or with so little help from conscious reasoning that I didn't notice it.

From the replies it seems there are two key realizations that combined could lead directly to the solution, and both realizations strike me as things that intuition would be good at noticing. The first is that each player likely has deterministic rule that is dependent on their coin, so there are only two candidate strategies per player. And the second is that the players likely have different strategies.

[meowface]

We had very similar thought processes.