Over history, people have found amazing solutions to the world's problems. Build the pyramids, put a man on the moon, send information securely over the internet to name a few. In the pure academic sense we've also solved incredibly tough problems like Fermat's Last Theorem and most of the Hilbert problems.
Long time ago as a budding mathematician I learned a few key principles of problem solving:
These principles can also help in analyzing poker situations. Often improving and understanding the game is met by road blocks because we don't pose the problem in the correct form. As a specific example consider consider the situation of playing 4-bet pots. Suppose you're playing a very aggressive and bad player. He 3-bets 50% of hands, rarely folds to 4-bets. I'm sure many hu players have played such opponents. It can be frustrating and not obvious how they are losing money. Let's pose the problem a little differently.
Since he defends (never flats) 50% of hands, we can't show an immediate profit by raising any 2. Thus, we obviously fold the weakest hands. It's not hard to see that that against such an opponent we should never raise/fold a hand (assuming a 3x open) since we don't shove immediate profit. Even air hands should be either folded or 4-bet. So say we pick some range, say 40% of 'playable' hands. I don't know what's optimal here, it depends on his postflop play, stack sizes, and our relative skill level.
Now pose the problem differently.
Hero (SB) posts an imaginary blind of 3bb (the open)
Villain (BB) posts an imaginary blind of 10bb (3-bet)
Flatting is equivalent to limping. 4-betting is equivalent to open raising. So we 4-bet some value range. With 100bb we have roughly a pot-stack-ratio of 1. We look at our equity against his calling range, his tendencies, and figure out the optimal play. I know you might hear complaints like it's hard to play AK when you miss the flop etc. But we are playing our 'opening' range against his 'flatting' range. If our range is better/stronger/more optimal than his we make money. Reducing the problem of 4-bet pots to a problem we understand better (how to beat opponents who flat too many weak hands oop), the situation isn't as daunting.
To recap, we reduce 4-bet pots against loose and non-folding opponents to a huge blinds game with a PSR of 1. This is an example of solving an analogous problem. The other techniques listed above also apply to poker. To be continued.