Let $w$ be a permutation of $\{1,2,\dots,n\}$ chosen uniformly at random. You have to determine $w$ by successively guessing permutations $v_1, v_2, \dots$. After each guess $v_j$ you are told where $v_j$ and $w$ agree, i.e., those values $i$ for which $v_j(i)= w(i)$. What is the optimal strategy to minimize the number of guesses? What is the expected number $E(n)$ of guesses? If an exact answer is intractable, can $E(n)$ be determined asymptotically? Is there a constant $0<c<1$ for which $E(n)\sim cn$?
Simple observation. Suppose that for $2\leq j\leq n$ we always choose $v_j$ to satisfy the following: if we already have determined that $w(i)=a$, then choose $v_j(i)=a$. If we have already determined that $w(i)\neq b$, then choose $v_j(i)\neq b$. (It seems plausible that some optimal strategy will have this property.) Such a choice is always possible by, for instance, cyclically shifting the previous incorrectly guessed values. Then using this procedure we will determine $w$ in at most $n$ guesses.
There are numerous related questions. Here are two.
(1) What if we only need to determine $w(1)$? If we are only allowed to guess a single number $v_j(1)$ each time, then the expected number of guesses is $(n+1)/2$. But we gain additional information when we guess the full permutation $v_j$, and this lowers the expectation.
(2) For $m<n$ what is the most (or least) number of ways that we can extend an $m\times n$ Latin rectangle, where $m<n$, to an $(m+1)\times n$ Latin rectangle (or to a Latin square)?
This guessing protocol is similar to that of the game Wordle. The protocol as described here (for $3\leq n\leq 7$) is an Event called Magic Cauldron of the video game Royal Match. Magic Cauldron is the motivation for the present posting.
