Another thing worth looking up in wikipedia is cryptographic hash
https://en.wikipedia.org/wiki/Cryptographic_hash_function
To solve the problems of Round 1, what Mr G valiantly attempted to design is essentially a cryptographic hash function. Round 1 began approximately like this:
Mr X: I have generated 4 random numbers between 1 and 50. Anyone is welcome to guess them.
Mr G: I guess 2, 12, 12 and 40.
Mr X: Your guess for the first number is correct.
There is a problem with this, which is that Mr X could have had a secret plan to convince us that telepathy is real when it is not, and he could have said that Mr G's first guess was correct
whatever Mr G had guessed.
There is also a mirror-image problem, which is that Mr X could have had a secret plan to convince us that telepathy does not exist even though it does, and he could have said that Mr G's guess was incorrect even if it was actually correct.
A nice way to solve both these problems is for Mr X to create for each secret number an object H, which has the following properties:
(1) it is easy for Mr X to create H and post it on the forum
(2) knowing H does not make it easier for anyone to guess the secret number
(3) anyone who makes a guess can easily ask H "is this number right or wrong?" and H will answer correctly.
This does open up the possibility that a determined guesser could guess every number between 1 and 50 in turn until he or she chances upon the correct number. But if we vastly increase the range of the random numbers, so that instead of being between 1 and 50 the secret number is between 1 and a million (for example) then it becomes infeasible to systematically try every number in turn.
So that is exactly what Mr G did. Noting that there are about a million words in English, he changed the game from a number-guessing game to a word-guessing game, and he showed how to create the object H with (hopefully) the properties listed above.