Does anyone have tips on how to glean and remember, long-term, the key concepts from tricky brainteaser/Olympiad-style problems (or tough problems in general) that aren't just memorizing the specific problem instance?
Context:
I was working on a problem (the famous "five-card" or Fitch Cheney trick), after having spent a couple of hours on it I began to wonder if I had seen a similar problem before. And after digging around, I found that I had actually solved this problem 11 months prior, and written up a brief solution sketch somewhere in my notes.
This was quite disheartening as I felt it meant that I had failed to really learn or take away any key concepts from the problem when I did it 11 months earlier. (To add--I was feeling completely stuck so finding that I had solved it then when I was unable to do so now was a double blow).
It isn't so much the specific solution as the general trick you want to remember.
Some tricks are basic, like completing the square, or simple symmetry, or "what is the chance of that NOT happening?". Some are a bit more involved, like the "stars and bars" argument for counting combinations.
It can help to have a very broad classification of tricks. For example, two of my classifications are "add zero" and "multiply by one". Completing the square is an example of "add zero". Changing units is an example of "multiply by one". Other tricks include:
"Break the problem into two problems" which has special cases such as "let $x = u + v$" where you are introducing two new variables to replace one. This gets used in solving the cubic, for example. Similarly "let Z(x,y) = X(x)Y(y)" when possible is factoring a problem, used in separation of variables.
For the trickier ones, it can help to have a list called, "I never would have thought of that." The specific tricks that seem to come out of the blue can sometimes be reused.
Another type of solution I call "the stars have aligned". If you have exactly the right number of items for a trick, for example. Many substitutions in integral calculus are of this form. You can keep an eye out for these unlikely approaches just in case the conditions work out just right.
Clearly there are more classifications as well. Good luck!