Tiling

Transcript

TilingCovering a region using a set of shapes, without gaps or overlaps, is called tiling. Let’s do an example 4 × 6 rectangular grid We call this a 4 × 6 grid, since it has 4 rows and 6 columns. Now, our question is Can a 4 × 6 grid be tiled using multiple copies of 2 × 1 tiles? Since we have 2 × 1 tile, we can use it horizontally or vertically One way to do it is More ways to do it can be Parity Parity basically means if its even or odd. And, we use that while tiling The Rule The most basic rule of tiling is simple arithmetic. If you want to cover a floor with tiles, the total area of the floor must match the total area of the tiles. For a 2 × 1 tile A domino (2 × 1) has an area of 2 unit squares. Therefore, any region tiled by dominoes must have an even number of squares (2, 4, 6, 8...). If the total number of squares is odd, tiling is impossible. Now, let’s answer the questions using Parity Can a 𝟒 × 𝟔 grid be tiled using 2 × 1 tiles? Total Squares: 4 × 6=24 Parity: Even. Answer: Yes. Can a 𝟒×𝟕 grid be tiled using 2 × 1 tiles? Total Squares: 4 × 7=28 Parity: Even. Answer: Yes. You can tile the first 6 columns easily, and the last column (4 × 1 ) can be filled with 2 vertical dominoes. One way could be What about a 𝟓 × 𝟕 grid using 2 × 1 tiles? Total Squares: 5 × 7=35. Parity: Odd. Answer: No. You cannot divide 35 squares into groups of 2 . There will always be one square left over. Let’s try using diagram We observe that one tile is left, so we cannot tile this are