Decide whether a grid can be covered with 2 × 1 tiles by counting squares, and use chessboard colouring to prove some regions impossible.
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Covering a region with tiles, with no gaps and no overlaps.
Covering a region with a set of shapes, without gaps or overlaps, is called tiling.
4 × 6, 4 × 7, 5 × 7, and the general rule.
Problem. Can a 4 × 6, a 4 × 7 and a 5 × 7 grid be tiled with 2 × 1 tiles? (Textbook pp. 157–158)
Decide for several grids.
Can a 6 × 9 grid be tiled with dominoes?
Colour the grid like a chessboard: each tile covers one black and one white.
A 5 × 3 grid with one square removed has 14 squares, an even number. Some such regions can be tiled, but some can't. How can we be sure one can't?
Colouring the 5 × 3 grid with its top-middle square removed.
Problem. Can a 5 × 3 grid with its top-middle square removed (Fig. 6.13) be tiled with dominoes? (Textbook pp. 159–160)
The regions from the book's Figure it Out, for the next task.
1. (Textbook p. 160) A region with rows of 2, 2, 4 and 4 squares, lined up on the left. Tile it with L-shaped tiles of 3 squares.
Two regions from the book: one with L-shaped tiles, one with dominoes.
Figure it Out, Textbook p. 160. Use the two regions shown above.
1. Tile the region with rows of 2, 2, 4 and 4 squares using L-shaped tiles made of 3 squares.
2. Tile the 8 × 8 board with its top-right and bottom-left corner squares removed, using 2 × 1 tiles.
Start with the top-left corner.
What colour are the two removed corners?