The system is designed such that a computer runs a cryptographic hash function, whose output cannot be predicted without running the function. What the computer does it uses the output of the first hash as an input of the second hash and so forth until you reach the nth hash. HashN represents the actual output. The computer only needs to publish a subset of the hashes and indices and, if each is correct, they only could have been computed by computing all other hashes before it, thus represproving that the work had been done.