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Here is a bit of advice that took me a while to learn:

You don't need to know what you are proving when you start to write a proof by induction.

The following method isn't needed for easy problems, but it has several times gotten me a lemma that wouldn't yield to any other means. After you've worked on the problem for a week or so and have a good feel for it, write down all the conditions that seem to be relevant. Write down all the implications you can prove of the form "If the conditions in set $S$ hold for some values of the parameters, then the conditions in set $T$ hold for some other values." Discard any in which set $S$ can be shrunk, or $T$ enlarged. Now, draw a graph with arrows between the sets. You are looking for a loop in this graph, a path from your hypothesis to the loop, and a path from the loop to your conclusion. If you find one, then you have a potential proof by induction, assuming that your parameters "decrease" as you go around the loop.

This is the other part of the trick. When you start this procedure, there is no need to decide which order you are using on the set of parameters. Wait until you've found the loop! Then the loop gives you a specific recursion on your parameter set, and you can try to find a well-order with respect to which this recursion is decreasing.

I'd been thinking about writing a blog post on this, but all the examples I know are really technical.

The following method isn't needed for easy problems, but it has several times gotten me a lemma that wouldn't yield to any other means. After you've worked on the problem for a week or so and have a good feel for it, write down all the conditions that seem to be relevant. Write down all the implications you can prove of the form "If the conditions in set $S$ hold for some values of the parameters, then the conditions in set $T$ hold for some other values." Discard any in which set $S$ can be shrunk, or $T$ enlarged. Now, draw a graph with arrows between the sets. You are looking for a loop in this graph, a path from your hypothesis to the loop, and a path from the loop to your conclusion. If you find one, then you have a potential proof by induction, assuming that your parameters "decrease" as you go around the loop.