Nonce

The nonce is a 32-bit field in the block header that miners change while searching for a valid block. "Nonce" is short for "number used once."

Mining at the lowest level looks like this:

loop:
  set nonce to next value
  hash = SHA-256(SHA-256(header))
  if hash < target: broadcast block
  else: try again

That's it. There's no shortcut, no algebra, no clever derivation. You just compute hashes until one happens to fall below the target. Modern ASICs do this around 100 trillion times per second per chip.

The 32-bit nonce only has 2^32 = ~4.3 billion possible values, which a serious mining operation burns through in a fraction of a second. When it runs out, the miner changes another part of the block (typically the extranonce inside the coinbase transaction), which changes the Merkle root in the header, which gives them a fresh 4.3-billion-value nonce space to search. Repeat until something works.

The nonce is the most boring 4 bytes in Bitcoin and also the entire mechanism by which proof-of-work happens. See Hash for what's being computed, and the Mining rabbit hole for why finding a good one matters.