Bitcoins are mined using a cryptographic algorithm called SHA-256. This algorithm is simple enough to be done with pencil and paper, as I show in this video. Not surprisingly, this is a thoroughly impractical way to mine. One round of the algorithm takes 16 minutes, 45 seconds which works out to a hash rate of 0.67 hashes per day.
For details, see http://righto.com/sha
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