42. An infinite grid sheet of paper is covered with a layer of rectangular tiles of size , with sides aligned along the grid lines. Prove that it can be covered with three more layers of tiles in the same manner so that no tile lies exactly on top of another.
Solution
83.42. We will lay the second layer as follows: divide the entire plane into 2x2 squares, each of which will be covered either by two horizontal or two vertical tiles. Then we will place the third layer as follows: connect with segments those adjacent cells that have not yet been covered by a single tile. Obviously, two segments come out of each cell. The lines formed by these segments are divided into two types: they are either closed lines of even length or infinite non-closed lines. Lines of both types can be divided into non-intersecting segments of unit length, corresponding to the tiles of the third layer. The fourth layer is laid uniquely: each cell is connected by a segment to exactly one adjacent cell.
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