Constructions and Tilings | IT

Question 41

Here is a $5 \times 3$ grid, with a unit square removed. Now, it has an even number of unit squares. Is it tileable with $2 \times 1$ tiles?

Solution

Understand the Question

Each $2 \times 1$ domino tile covers exactly $2$ adjacent squares.
When a grid is colored alternately like a chessboard (black and white), every $2 \times 1$ tile must cover exactly $1$ black square and $1$ white square.
Therefore, for any board to be tileable by $2 \times 1$ tiles, the number of black squares must equal the number of white squares.
We use chessboard coloring to count the remaining black and white squares and check if they are equal.

Step 1 · Count the Remaining Squares

Total squares in a $5 \times 3$ grid: $5 \times 3 = 15$

After removing one unit square: $15 - 1 = 14$ Diagram 1

Since each $2 \times 1$ tile covers $2$ squares, the remaining $14$ squares would require $\dfrac{14}{2} = 7$ tiles.

Step 2 · Analyze Using Chessboard Coloring

Color the $5 \times 3$ grid with alternating Black ( $\text{B}$ ) and White ( $\text{W}$ ) squares:

\begin{array}{|c|c|c|} \hline \text{B} & \text{W} & \text{B} \\ \hline \text{W} & \text{B} & \text{W} \\ \hline \text{B} & \text{W} & \text{B} \\ \hline \text{W} & \text{B} & \text{W} \\ \hline \text{B} & \text{W} & \text{B} \\ \hline \end{array}

Total squares by color in the full grid:

Black squares: $2 + 1 + 2 + 1 + 2 = 8$
White squares: $1 + 2 + 1 + 2 + 1 = 7$

The removed square is the center square (row 3, column 2), which is $\text{White}$ : $\text{Remaining White squares} = 7 - 1 = 6$ $\text{Remaining Black squares} = 8$

Since each $2 \times 1$ tile covers exactly $1$ black and $1$ white square, a tileable board requires an equal number of black and white squares. Here, $8 \neq 6$ , so tiling is impossible.

Answer

No, the grid is not tileable with $2 \times 1$ tiles.

Common Mistakes

Assuming Even Count Is Sufficient: Having an even number of total squares ( $14$ ) is necessary, but not sufficient on its own. Color parity (equal black and white squares) must also hold.
Misidentifying Removed Square Color: Double-check whether the removed square corresponds to a black or white cell in the chosen coloring pattern.

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Q28

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Q. How do we do it?

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Why is $\angle \text{CAX} = 60^\circ$ ? Is there an equilateral triangle here?

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Is an $m \times n$ grid tileable with $2 \times 1$ tiles, if both $m$ and $n$ are even? If yes, come up with a general strategy to tile it.

Q39

Is an $m \times n$ grid tileable with $2 \times 1$ tiles, if one of $m$ and $n$ is even and the other is odd? If yes, come up with a general strategy to tile it.

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Q41

Here is a $5 \times 3$ grid, with a unit square removed. Now, it has an even number of unit squares. Is it tileable with $2 \times 1$ tiles?

Q42

Is the following region tileable with $2 \times 1$ tiles?

Q43

Context: We are considering whether a given region can be tiled using $2 \times 1$ tiles.

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Q44

Were you able to tile this? How can we be sure that this is not tileable? Can you find another unit square that, when removed from a $5 \times 3$ grid, makes it non-tileable?

Q45

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Q46

If the black-and-white grid is tileable, is the plain grid tileable?

Q47

Use this idea to find another unit square that, when removed from a $5 \times 3$ grid, makes it non-tileable?