I recently caught the flu double header. As appropriate for someone in my condition, I spent a good many hours riding around on city buses, mumbling to myself and reading about bitcoin on my phone. If you are looking for a decent semi-technical introduction, Michael Nielsen's explanation is recommended.
The part of bitcoin that strikes me as bizarre is what the proof-of-work exercise entails. Essentially, to sustain an agreed-upon but decentralized public record of transactions, participants are madly trying to solve a useless reverse-hashing puzzle. Basically, "guess some bits such that when you append them to this fixed long string of bits, the hash starts with at least 5 (or whatever) zeroes." By making the puzzle hard to solve and easy to verify, and rewarding those who solve it, the system has accountability and resilience, and is robust against takeover.
However, it is hard not to see the hashing puzzle as a satire of contemporary work culture: participants are paid to use their computers to solve numeric puzzles which are of no interest to anyone. (Never mind the potential environmental impact if cryptocurrencies see greater adoption.)
You know who else liked Ansatz?
However, the hashing puzzle reminded me of something, in my feverish state. In the first weeks of differential equations classes, it is customary to pose a differential equation, then present the solution, deus ex machina, and confirm it is the solution. Hard to solve, easy to verify. Partial differential equations have the same nature. For example, the solution to the heat equation involves some drudgery, but confirmation of the solution is pretty simple. In fact, computers can even verify solutions of differential equations because symbolic differentiation is relatively simple.
So what if we could make an altcoin where the proof of work involved the solution to a real-world math problem? Suppose, for example, that SETI could pay some small fee to have altcoin users comb through radio data in the search for alien life, while maintaining the ledger of transactions. If it sounds too tin-foil hat to have cryptocurrency looking for ET, consider the impact that Captcha has had by taking two needs and connecting them together at large scale. Maybe the one-sided proof-of-work puzzle of altcoins can be matched to somebody else's computational needs.