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Hey, I've just started learning DSA and I am coming across different types of problems that are considered to be classics like Tower of Hanoi, Josephus, Subset Sum, Nqueen's etc

My question is, why do we have to study these problems? what do we gain? how do we apply this understanding?
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If your goal is to be able to figure out and write down an algorithm to solve a problem you haven't seen before, you will want to get lots of practice doing exactly that. The problems you listed provide great examples of how to break a problem down into sub-problems, and by working through them, you'll see that certain patterns in sub-problems tend to show up all over the place in different problems. You learn general problem-solving methods.

Then, when you need an algorithm to analyze an audio signal for the DSP course in a couple years, or whatever, it will hopefully occur to you to break your signal up into subsignals and go about solving it that way.
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If they gave you problems without a known solution... They wouldn't even know if they are solvable at all, and you wouldn't be able to improve or get help about it
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Consider it as greater, abstract patterns that are common to many processing tasks. Recognizing those patterns helps you to break that task down with tools.
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