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Does the amount of time left determine how many points I get or does it not matter as long as it hasn't expired yet?
I do believe that the more time you spend solving a tactic, the less points you get. (even negative points for taking too long)
Yeah, and it's silly IMO. But that's why I prefer chess tempo, despite being less polished.
Part of the training is to sharpen your tactical eye so you recognize the patterns quickly, which translates into a useful skill in a real game at any time limit.
Solving puzzles in itself is good practice and builds your skill, but you can't take unlimited time looking for tactics when your clock is ticking!
I just did one that started with 20+ min which seemed like overkill but I only got 8 points after doing it in one minute with a rating
It also depends on the rating of the problem vs your rating.