The big problem is that there isn't enough space on all the computers in the universe to store the equation for just one of the bits, even if fully simplified. That and even if every particle in the universe were a supercomputer, the age of the universe wouldn't be long enough to solve the equation. The equations are not linear and so they don't simplify very much.
Cryptographers know of this symbolic reduction technique - and SHA-256 and AES and RSA and about eveyr single ASIC design process uses SymRed to optimize their circuits.
Let's let the OP try to write up the logic formula for SHA256's first 32 bits of input and 1 bit of output. Then try to use Karnaugh Maps (
http://en.wikipedia.org/wiki/Karnaugh_map) or other technique to simplify it. I predict it will educate them on how hard a problem this is and what kinds of things cryptographers think of when designing crypto functions.
Edit: Also look at:
http://en.wikipedia.org/wiki/Circuit_minimization