Area 51

As ${1 \le h, w \le n}$, there are total $n^2$ possible area ${(1, 2, 3, \dots, n^2)}$. We can split it into sequential $n$ segments where length of each segment is $n$ {${(1, \dots, n), (n+1, \dots, 2n), (2n+1, \dots, 3n), \dots, (n(n-1)+1, \dots, n^2)}$}.

At first, query on these segments and find secret area lies in which segment. It can be done with binary search on $1st$ segment to $n−th$ segment. It takes at most $10$ queries.

Now, we’ve a $n$ length segment means $n$ possible area. Iterate on each area, take only valid area. Valid area means which integer has at least two divisors (divisors can be same) less than or equal to $n$. Now binary search on valid area. It takes at most $10$ queries.