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That's not an accurate way to look at it. There is no "part of the puzzle". Only "the puzzle". The name of the game is to get a hash that starts with a some number of zeros (higher difficulty -> more zeros). In a pool, each participant sends in answers to the puzzle that are of some much easier difficulty. This allows the pool to gauge how much work they are doing and give out rewards in accordance with that.

Thanks wheatstone!

This is exactly what I understood about mining so far.
There is no part of the puzzle, only THE puzzle. To pick up the reward you have to come up with low enough hash.

I was wondering how does a pool split this computationally difficult task into smaller tasks while at the same time preventing any individual miner to claim the reward if he happens to be the one hitting the correct solution?

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There are many ways to do pool mining I'll give an intro to the stratum protocol as it's the most used.
With the stratum protocol each worker is given an ExtraNonce1, which is essentially the workers id each worker will have a different ExtraNonce1, and the size of ExtraNonce2, the number of bytes the worker can iterate. Example ExtraNonce1 "08000002" and ExtraNonce2 4. Each worker is also given enough information to build a unique block header: hash of previous block, coinbase1 script, coinbase2 script, list of merkle branches, block version, bits, and time.
Coinbase1 + ExtraNonce1 + ExtraNonce2 + coinbase2 = the coinbase transaction, this is the transaction where the miners reward goes to. The workers receive coinbase1 and coinbase2 they can not change it.
Once the the coinbase transaction is created it is combined with the merkle branches to create a merkle root. At this point the worker has all the components to create a header which can be worked on by incrementing the standard nonce.

Thanks GoldenWings91

I need some time to digest this, some of the terms are unfamiliar to me so I'll get back to reading some more.

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Solominers and pools can see the transactions and select the ones they wish to include. They are completely free to chose which and how many to include.

Then what incentive do miners have to include zero-fee transactions?
Can it happen that such transactions float forever without being included? Are they dropped after some time?