Question 113
Let $m, n$ be positive integers. $m \times n$ square boxes of side length 1 form a grid for a rectangle (or a square if $m=n$) of sides $m$ and $n$. We want to color each of the $m \times n$ boxes by using one of the colors red, blue or black in such a way that all of the following conditions are satisfied:
* every red box is adjacent to exactly 1 blue box and exactly 1 black box.
* every blue box is adjacent to exactly 1 black box and exactly 1 red box.
* every black box is adjacent to exactly 1 red box and exactly 1 blue box.
Here, we say that two different boxes are adjacent to each other if they have a side in common.
Answer the following questions in this context:
(1) Give an example of coloring satisfying the conditions given above in case $m=3, n=4$. (Give just an example.)
(2) Determine all the pairs $(m, n)$ for which the coloring of boxes satisfying the conditions given above is possible.
* every red box is adjacent to exactly 1 blue box and exactly 1 black box.
* every blue box is adjacent to exactly 1 black box and exactly 1 red box.
* every black box is adjacent to exactly 1 red box and exactly 1 blue box.
Here, we say that two different boxes are adjacent to each other if they have a side in common.
Answer the following questions in this context:
(1) Give an example of coloring satisfying the conditions given above in case $m=3, n=4$. (Give just an example.)
(2) Determine all the pairs $(m, n)$ for which the coloring of boxes satisfying the conditions given above is possible.