Proof of Work, in Bitcoin context:
Find a nonce (number used once) that makes the output of SHA256(SHA256(block header + nonce)) smaller than a certain number (difficulty).

The question is if there is any way around simply trying nonce = 0, nonce = 1... and so on until you find a fitting one, or if there are shortcuts or strategies to find one quicker.
The OP seems to be trolling though and I love the fact that for once this is being ignored and turned into an actually nice and interesting thread!

Lookup tables and so on might be used in the second iteration, you could store SHA256 hashes of the first round and if they lead to a hash that is already known in th secod round, you don't need to calculate that one. I doubt though that there would be many benefits from that, as the number of hashes starting with 128 zeros (that's a VERY high difficulty!) is still 2^128... (that's a VERY high number of hashes to store). You'll waste far more energy doing the (in most - with most meaning all - cases unsuccessful) lookup before doing the calculation of the second hash anyways.