So it's mostly useful for problems of moderate solvability, that you could solve without the power, given a considerable effort (but not more effort than you would be willing to actually put forth).

For example, you could set Hashcash to mint some expensive stamp, and firmly decide to check its output in three days.

(For those who don't want to take the time to read the link, Hashcash is a proof-of-work system based on brute-force partial preimage attacks against hash (one-way) algorithms.)