Let $m, n > 2$ be fixed integers. $A$ thinks of $n$ not necessarily distinct integers, $x_1, x_2, ..., x_n$, such that $1 \le x_i \le m$ for all $1 \le i \le n$. For each query, $B$ chooses and tells $A$ $2n$ integers, $a_i,b_i (1 \le i \le n)$, such that $1 \le a_i \le b_i \le m$ for each $i$. (Note that $a_i$ can be equal to $b_i$ for some $i$.)
$A$ replies "Bounded" if $a_i \le x_i \le b_i$ for ALL $i$ in the range $[1,n]$. Otherwise, even if it is not true for just one value of $i$, $A$ replies "Out of Bound".
Find the minimum number of queries $B$ needs to guess all of $A$'s numbers correctly.
This is a question I thought of myself. I asked a similar question before, where $n=3$ and $m=10$.