Tiling

The idea

Covering a region using a set of shapes, without gaps or overlaps, is called tiling.

What is a tiling?

A floor is covered with tiles. A good floor has no empty patches and no tiles lying on top of each other.

A region is tiled when the tiles:

• cover all of it with no gaps and no overlaps
• cover most of it
• cover all of it, and may overlap

Tile the grid

Tap a square, then a square next to it: a 2 × 1 tile covers both. Tap a tile to take it away.

Count the squares

Every tile covers 2 squares. The 4 × 3 grid has 12 squares, so it needs 12 ÷ 2 = 6 tiles. The 3 × 3 grid has 9 squares, an odd number. Tiles cover squares two at a time, so one square is always left over.

Tiling a region means covering it with tiles with no gaps and no overlaps. With 2 × 1 tiles, a region with an odd number of squares can never be tiled.

• Grid | Squares | Tiled?
• 4 × 3 | 12 | yes, with 6 tiles
• 3 × 3 | 9 | no, 4 tiles and 1 square left

Notes

Tiling a region means covering it with tiles with no gaps and no overlaps. With 2 × 1 tiles, a region with an odd number of squares can never be tiled.

Check yourself

A 4 × 7 grid has to be tiled with 2 × 1 tiles. How many tiles are needed?

Answer: 14

The grid has 4 × 7 = 28 squares and each tile covers 2, so 28 ÷ 2 = 14 tiles.

Can a 5 × 7 grid be tiled with 2 × 1 tiles?

Ria removes one corner square from a 5 × 3 grid. The rest can be tiled. How many 2 × 1 tiles does it need?

Answer: 7

5 × 3 = 15 squares, minus 1 leaves 14, and 14 ÷ 2 = 7 tiles.

• Yes, if we turn some tiles sideways.. Turning tiles does not change the count: each covers 2 squares, and 35 is odd.
• No. It has 35 squares, an odd number, and tiles cover two at a time. — correct. Yes! One square would always be left.
• Yes, with 17 tiles and a gap that is too small to see.. 17 tiles cover 34 squares, so a square is left. That is a gap.