Question 61
Let $m$ and $n$ be two integers greater than $3$ and let be given a rectangular $m \times n$ board. At each step, one puts simultaneously $4$ marbles into $4$ cells of the board (each marble into a cell) so that these four cells form one of the following schemata.
Is it true that starting from a rectangular $m \times n$ board without marbles in it after a finite number of steps of appropriate puttings, one can put marbles into all cells of the board so that each cell is filled with a same (positive) number of marbles for all cells when:
i) $m = 2004$ and $n = 2006$?
ii) $m = 2005$ and $n = 2006$?
(At each step, it is not necessary that the four cells which are selected to put marbles into contained no marbles).
Is it true that starting from a rectangular $m \times n$ board without marbles in it after a finite number of steps of appropriate puttings, one can put marbles into all cells of the board so that each cell is filled with a same (positive) number of marbles for all cells when:
i) $m = 2004$ and $n = 2006$?
ii) $m = 2005$ and $n = 2006$?
(At each step, it is not necessary that the four cells which are selected to put marbles into contained no marbles).