Constructions and Tilings · Lesson 9 of 12
Tiling Rectangular Grids
“Discover exactly which rectangular grids can be tiled by two-square dominoes and explain the general strategy.”
• Interpret a grid’s row and column counts. • Place a 2 × 1 tile in either orientation. • Use an area count to rule out odd-sized regions. • Construct tilings when one rectangle dimension is even. • State and justify the criterion for an intact rectangular grid.
Rows, Columns, and Unit Squares
A rectangular grid makes the covered area easy to count. Rows run across the grid and columns run down it. A 4 × 6 grid has four rows and six columns, so each of its four rows contains six unit squares.
A square used as one unit of area in the grid. A grid with m rows and n columns contains mn unit squares.
Horizontal and Vertical Tiles
Our tile is a 2 × 1 rectangle, often called a domino. It covers exactly two adjacent unit squares. Rotation is allowed, so the two squares may be side by side in a row or one above the other in a column.
A domino cannot cover two separated squares or two squares that touch only at a corner. Its two unit squares share a full edge. In this grid problem we align tiles with the grid so that each placement covers two whole cells. Turning a horizontal tile through a right angle gives a vertical tile without changing its area.
If a tiling exists, the number of tiles is half the square count. That count is necessary but does not tell us where to put them. We must still show an arrangement that covers every cell exactly once.
Tiling 4 × 6 and 4 × 7 Grids
Begin with the source’s two possible rectangles. A 4 × 6 grid has an even number of rows and columns; a 4 × 7 grid has only an even row count. A vertical-column strategy works for both because every column contains four squares.
Problem
How many dominoes are needed, and how can they be arranged?
- 1.The square count is 4 × 6 = 24. A domino covers two squares, so 24 ÷ 2 = 12 dominoes are needed.
- 2.Pair rows 1 and 2, and pair rows 3 and 4 within every column.
- 3.Each column now contains two vertical dominoes; six columns use 6 × 2 = 12.
- 4.Every square belongs to one of these pairs. This is an explicit tiling, not merely an even-area calculation.
Problem
Can a 4 × 7 grid be tiled?
- 1.It has 4 × 7 = 28 squares, requiring 28 ÷ 2 = 14 dominoes if a tiling is possible.
- 2.The four rows can be paired vertically within each column.
- 3.Place two dominoes in each of the seven columns, giving 7 × 2 = 14 dominoes.
- 4.The odd column count causes no difficulty because the even row count provides the pairing strategy.
Why a 5 × 7 Grid Cannot Be Tiled
Sometimes a count proves that no arrangement can work. A 5 × 7 rectangle has 35 squares, while every domino covers an even number of squares. Any number of dominoes therefore covers an even total, never exactly 35.
Problem
Explain why trying many placements cannot solve a 5 × 7 domino tiling.
- 1.The region has 5 × 7 = 35 unit squares.
- 2.If k dominoes covered it, they would cover 2k squares. But 2k is even for every whole number k.
- 3.The required count 35 is odd, so 2k = 35 has no whole-number solution.
- 4.At least one square must remain unpaired. Changing orientations does not change the number each tile covers.
General Strategies for an m × n Grid
The examples suggest a rule, but a general statement needs both a construction for the possible cases and a reason for the impossible case. There are three parity cases to consider: both dimensions even, one even and one odd, or both odd.
| Dimensions | Strategy or obstruction | Conclusion |
|---|---|---|
| Both even | Pair vertically within each column, or horizontally within each row | Tileable |
| Rows even, columns odd | Pair vertically within every column | Tileable |
| Rows odd, columns even | Pair horizontally within every row | Tileable |
| Both odd | The product mn is odd, so no pairing covers all cells | Not tileable |
If m is even, divide each column into pairs of consecutive cells. Because no cell is left over, vertical dominoes cover every column. If n is even, divide each row into pairs and use horizontal dominoes. If both m and n are odd, their product is odd, so a domino tiling is impossible.
Thus an intact rectangular grid is tileable exactly when at least one dimension is even. Equivalently, its area is even. The word intact matters: removing squares creates a different region, for which an even area alone may no longer guarantee a tiling. We will investigate that next.
Do not conclude that “odd columns means impossible.” A 4 × 7 grid has odd columns but even rows and is tileable. Also, the rectangular sufficiency rule does not automatically apply to rectangles with holes or removed cells.
Find another 4 × 6 tiling that combines horizontal and vertical dominoes. Check that each unit square appears in exactly one tile. More than one arrangement can satisfy the same coverage rules.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
How many unit squares are in a 4 × 7 grid?
Which two squares can one grid domino cover?
Why is a 5 × 7 grid impossible to tile by dominoes?
Which strategy tiles a 7 × 4 rectangle?
What is the complete criterion for an intact rectangle?
Practice Problems
- Draw and tile a 4 × 6 grid in two different ways.
- Tile a 4 × 7 grid and count the dominoes.
- Decide whether 7 × 8 and 9 × 5 grids are tileable. Justify both.
- How many dominoes would a valid 12 × 3 rectangle tiling use? Give a placement strategy.
- Explain the general strategy when m is odd and n is even.
- A student says every even-area region can be tiled. What limitation must be added to the rule from this lesson?
One arrangement uses vertical pairs in every column; another uses horizontal pairs in every row. Mixed arrangements may work too if every cell is covered once.
Key Takeaways
• An m × n grid has mn unit squares. • A domino covers two cells sharing a full edge. • A tileable region uses half as many dominoes as unit squares. • Even rows permit vertical pairing; even columns permit horizontal pairing. • An intact rectangle is tileable exactly when at least one dimension is even. • Odd area proves impossibility, but even area alone is not a universal test for other shapes.