I've noticed the recently solved block hashes have started with a number of zeros (8 last time I checked). Couldn't we just tell our mining hardware/software to assume the first few digits are in fact zeros and give us a head start?
All block hashes start with a certain number of zeroes by design. The nature of a hash is that knowing something about the output does not help you figure out what the input is supposed to be (at least in theory). So in short, no.