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Is the winning hash the same as the nonce?
No.
Is the miner is trying to solve the nonce or a winning hash?
Nonces are not solved. Talk of winning is a cause of confusion. This and talk of solving puzzles are either poetry or lies told to children. We must climb one rung in Wittgenstein's ladder.
Miners construct a block from valid data, then compute the hash of that block.
If the hash is numerically less than the current target, the block is valid. The miner stops mining and publishes the block. Then they resume mining with a completely new block.
If the hash is not less than the target, the block is not valid. The miner changes something in the block and starts again at step 1. The thing they change first is a value known as the nonce (which stands for number used once) which is expressly provided for this purpose in the block structure. After trying all possible nonce values, they can alter something else, like the timestamp or the choice and order of transactions etc.
As you take a step up Wittgenstein's ladder, you must completely discard the lower rung your foot just stepped off. This website allows people who are one step ahead to guide your foot. Clinging to the lower rungs will impede your progress.
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