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It's pretty hard to do. It appears that you need problems where (1) you can generate a practically unlimited number of arbitrary instances, (2) of arbitrary difficulty, (3) deterministically based on a random seed, (4) that are roughly constant difficulty to validate, and (5) where a randomly-chosen instance is likely to be useful to know a solution to.

If someone thought those cycles could be put to better use they would pay out bitcoins themselves to mining machines to convince them to run their computation instead, it's straight arbitrage.

You might be able to convince a lot of cryptocurrency adopters to switch to coins that were based on a useful proof-of-work (sometimes called "proof-of-useful-work") even if the kind of useful work that secures the coin isn't one that many people are currently paying for. After all, some present and prospective adopters care a lot about externalities, whether ethically or out of fear of future regulation or marketing.

A wonderful case would be where biomedical research somehow had an unlimited and ongoing need for solutions to arbitrary, random simulation tasks (where it was also relatively easy to confirm the correctness of an alleged solution). This doesn't really seem to be the case, but cryptocurrencies' impact and popularity would be improved quite a bit if it were!

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