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Ok, it's a yes or no question: miners can include different transactions in a block, so does that mean that they search for different nonces

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Yes, different miners will almost certainly always search for different nonces. Besides including different transactions from the network, they will also almost certainly send the proceeds from the coinbase transaction to different accounts. There are many other places where the block can also be different.

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    In fact at current difficulty the algorithm isn't, as often presented in ancient tutorials, 'search for a nonce that makes an otherwise-fixed block valid' but rather 'search for a block (by varying at least coinbase) for which there exists any nonce that makes it valid'. – dave_thompson_085 May 9 at 5:58

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