TITLE: Square board game with overlapping square sub-parts
QUESTION [7 upvotes]: Two players, A and B play a game on a board with $NxN$ squares. Player A populates all squares with numbers from 1 to $N^2$ in a completely random way. Here is an example for a 5x5 board:

Player B does not see the board but can ask player A to tell him the content of any square part of the board. Two such square parts (with red and green frame) are shown here:

Player A has to report numbers in the requested square part but he does not have to report them in any particular order. For example, for the red part of the board he could respond with: 23, 21, 12, 4, 14, 10, 5, 22, 25. And for the green part he could reply with: 15, 21, 11, 1.
How many questions player B has to ask before he is able to fully reconstruct the content of the board?
Obviously he can do it by asking for a content of every single 1x1 square of the board. So in the worst case he does not have to ask more than $N^2-1$ questions. But with a better strategy that number can be greatly reduced.

REPLY [3 votes]: For $N=2k+1$ there is a solution with $4k$ questions. 
Let $x_1,\dots,x_{N^2}$ be the set of all $1\times1$ squares, and let $S_1,\dots,S_{4k}$ be all of the $(k+1)\times(k+1)$ squares which touch an edge of the board. It is easy to see that, if $x$ and $y$ are any two different $1\times1$ squares, there is some $S_i$ which contains just one of them. In other words, the map $x\mapsto\{S_i:x\subset S_i\}$ is an injection from $\{x_1,\dots,x_{N^2}\}$ to $\mathcal P(\{S_1,\dots,S_{4k}\}).$ Therefore, asking for the contents of each of the squares $S_1,\dots,S_{4k}$ will enable us to reconstruct the board.
If $N=2k$ there is a solution with $4k-1$ questions, namely, we use all of the $k\times k$ squares which touch an edge of the board, except for one corner square.