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For example, in the Genesis block in Bitcoin, from which values was the solution hash "000000000019d6689c085ae165831e934ff763ae46a2a6c172b3f1b60a8ce26f" derived from? I understand that the nonce is applied to something to get the solution, but I can't seem to find what those values are.

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The process of mining consists on constantly hashing the block header in order to obtain a hash lower than a specific target. The block header contains (from the bitcoin wiki):

  • Version
  • hashPrevBlock
  • hashMerkleRoot
  • Time
  • Bits
  • Nonce

So when the obtained hash does not meet the required conditions, the nonce value is increased by one, so that the hash of the block header changes. The miner keeps changing the nonce, trying to find a value that makes the hash of the header lower than the current target.

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