Question 43
The numbers $1, 2, \ldots, 10000$ are put in some order into the squares of a checkered $100 \times 100$ board, one number per square. Pete marks the squares according to the following rules. At the beginning, he just marks $k$ squares by his own choice. By any subsequent move, he may mark any unmarked square containing a number $a$, if either (i) its row contains an already marked square with a number smaller than $a$, or (ii) its column contains an already marked square with a number greater than $a$.
Find the least $k$ such that for any arrangement of the numbers Pete is able to mark all the squares of the table. (S. Berlov)
Find the least $k$ such that for any arrangement of the numbers Pete is able to mark all the squares of the table. (S. Berlov)