TITLE: What is the minimum number of guesses in order to guarantee to win the prize?
QUESTION [7 upvotes]: Your friend will pick a $4$-letter word and you will make guesses in order to find it.
-A word can contain only the letters $A, B, C,\:\text {and} \:D$, and they can be used more than once. $(AAAA-DDDD)$.
-In your guess if at least three letters are in their correct places you will win a prize.
What is the minimum number of guesses in order to guarantee to win the prize?

REPLY [5 votes]: You are looking for the covering code number $K_4(4,1)$. Finding such numbers are very hard problems in general.
This particular one can be found in this paper, and it is equal to 24.
The covering table is also given, I reproduce it here:
AAAA
AABB
ABAB
ABBA
ACCC
ADDD
CACD
CCDA
CDAC
DDCA
DADC
DCAD
BAAB
BABA
BBAA
BBBB
BCCC
BDDD
CBCD
CCDB
CDBC
DDCB
DBDC
DCBD
Reference: Covering theorems for vectors with special reference to the case of four and five components.
R. G. Stanton, J. D. Horton and J. G. Kalbfleisch