That's not the idea. The computational problem is simply a proof of work--you create a specific, verifiable string, with a changeable component greater than 256 bits, then hash that string using incrementing values until you find a SHAsum less than a certain value. Knowing a random, short preimage for each possible hash doesn't help you there, because those will not allow you to create a verifiable string with the correct hash.
Now you can donate to Singularity Institute using Bitcoin.
Currently Bitcoin mining appears to be profitable as only bubble economics can be. Already some Less Wrong users have purchased GPUs and started mining Bitcoin. Please consider sending some Bitcoins to SI at address 1HUrNJfVFwQkbuMXwiPxSQcpyr3ktn1wc9
2014 Edit: Please donate Bitcoin by using the Bitpay link on MIRI's donate page. Thanks!