A rectangular floor is covered entirely by 2×2 and 1×4 tiles. One tile accidentally gets completely broken.
Each of the available spare tiles are corresponds to the other type.
Can that the floor be completely covered by rearranging the tiles?
Justify your answer.
*Assume that there are an unlimited number of spare tiles.
Perhaps there is a minimum width and length for a rectangular room such that above that critical value you can assuredly find a rearrangement.
Any such rearrangement must of necessity involve replacing not only the broken tile, but also some other tiles as well. This is shown by imagining the floor to be composed of 1x1 squares like a checkerboard, but instead of having only 2 colors, it has 4 colors:
In that pattern, the 2x2 pieces necessarily cover one of each color. The 1x4 pieces cover 2 of each of 2 colors. So replacing one tile with one tile won't do.
But again, since we have an unlimited supply of the "wrong" type tile, does a sufficiently large space allow multiple tile substitution to save the day? Clearly if the area is too narrow, you can't use 2x2's at all. Is there a minimum width and depth which will always allow these multiple substitutions?
Posted by Charlie
on 2015-03-20 11:00:28