A few thoughts about my own experience with the slinky problem.
I recall that it took me something like an hour to figure out that the video is not a trick and what the underlying physics was. I could be wrong on the time frame, though, as it was about a year ago.
I did not write any differential equations, however, just had to make the connection between the fact that the bottom was not moving and the conclusion that the top was essentially a shock wave. After that all that was left is to verify that the longitudinal slinky waves (not sound waves) are indeed slow enough for the shock to form very quickly.
Now, had I read the Polya book and applied the mantra, would I have found the solution any faster? I doubt that, though there is no way to check. I was quite familiar with shock waves already, as well as with the pole-in-the-barn paradox, having had to explain the latter on IRC many times. However, I did not pay attention to the fact that the contraction of the pole after its front hits the gate is due to a shock wave, not a sound wave, the latter being much too slow in relativistic circumstances. Why did I miss that? Probably because it was not essential to resolving the paradox, as once you realize that there are no rigid bodies in relativity, the paradox goes away.
I suspect that I internalized the "have you seen a similar problem before?" approach as much as I could already, but not expecting to see shock waves on such a slow time scale delayed the realization significantly.
The other day at dinner, someone showed me this video of a slinky dropping. It shoes that the bottom of the slinky stays perfectly stationary for a while after it's been dropped. (The link goes to the 10-second interesting part).
I spent some time trying to figure out why that happens, but didn't get it. The next day, I spent half an hour writing down the differential equations that describe the slinky's motion and staring at them, with no idea how to proceed. Eventually, I watched the video again with sound, and learned the simple answer, which is that the speed of waves traveling in a slinky is very slow - a few meters per second - and the bottom half sits still until a wave can travel down and inform it that the slinky's been dropped.
The strange thing is that I already knew this, or at least the idea was familiar to me. Also, while at dinner, someone mentioned the "pole-in-the-barn" paradox from special relativity, and mentioned the same speed-of-information-in-materials idea in resolving the paradox, but I still didn't make the connection to the problem I was considering.
I want a simple phrase, similar to "check consequentialism", "take the outside view", or "worth it?" that applies to checking your own thought process while solving problems to stop you from revving your engine in the wrong direction for too long. I realized I've read a book about what to do in such situations. It's George Polya's How to Solve It. (Amazon Wikipedia Google Books) I don't have a copy of the book anymore, and I would like to crowdsource creating a short phrase that captures the general mindset endorsed by it. Some questions I remember the book suggesting are