Question 4
(1) Player $A$ writes down two rows of $10$ positive integers, one under the other. The numbers must be chosen so that if $a$ is under $b$ and $c$ is under $d$, then $a + d = b + c$. Player $B$ is allowed to ask for the identity of the number in row $i$, column $j$. How many questions must he ask to be sure of determining all the numbers?
(2) An $m \times n$ array of positive integers is written on the blackboard. It has the property that for any four numbers $a$, $b$, $c$, $d$ with $a$ and $b$ in the same row, $c$ and $d$ in the same row, $a$ above $c$ (in the same column) and $b$ above $d$ (in the same column) we have $a + d = b + c$. If some numbers are wiped off, how many must be left for the table to be accurately restored?
(2) An $m \times n$ array of positive integers is written on the blackboard. It has the property that for any four numbers $a$, $b$, $c$, $d$ with $a$ and $b$ in the same row, $c$ and $d$ in the same row, $a$ above $c$ (in the same column) and $b$ above $d$ (in the same column) we have $a + d = b + c$. If some numbers are wiped off, how many must be left for the table to be accurately restored?