Bob: Hey Alice, lets play "Guess a number".
Alice: Oh I couldn't be bothered...
Bob: Oh come on, please?
Alice: Fine, I'll play. But you must submit your guesses in batches of 4. Except at the end. On your last attempt you can guess just one number, but if you are wrong you lose.
Bob: Oh, OK. But what will you tell me about my guesses when I make 4 at once?
Alice: The usual. Whether each one is too small, too big, or correct. Tell you what, I'll make it a bit easier. I'll choose a number between 1 and 6249.
So Alice thinks of a number between 1 and 6249.
Bob may play as many rounds of guessing four numbers as he wishes. [Note: the four guesses need not be distinct, but they will count for four guesses]. On the final round he must commit to a single number that he believes is Alice's number.
How many guesses does Bob need in order to correctly identify Alice's number and guarantee a win?
Remember, your answer must be 1 mod 4.
Follow up to this problem
From Guessing games