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

Question

Homi · Accepted Answer

Understand the Question

A standard domino tile ( $2 \times 1$ ) covers exactly two adjacent squares: one black and one white on a checkerboard-colored grid.
Therefore, a grid is tileable by dominoes only if it has an equal number of black and white squares.
The plain grid and the black-and-white grid represent the exact same shape. If the black-and-white grid is tileable, the plain grid is also tileable.

Step 1 · Count Total Squares in the Region

A full $3 \times 5$ rectangular grid has: $3 \times 5 = 15 \text{ squares}$ Diagram 1

Subtracting the $1$ missing square from the top middle: $15 - 1 = 14 \text{ squares}$

Step 2 · Count Black and White Squares

Diagram 2

Counting the squares by row:

Row 1: $2$ white squares, $0$ black squares
Row 2: $1$ white square, $2$ black squares
Row 3: $2$ white squares, $1$ black square
Row 4: $1$ white square, $2$ black squares
Row 5: $2$ white squares, $1$ black square

Total white squares: $2 + 1 + 2 + 1 + 2 = 8 \text{ white squares}$

Total black squares: $0 + 2 + 1 + 2 + 1 = 6 \text{ black squares}$

Step 3 · Check Tileability Condition

Each domino tile covers exactly $1$ white square and $1$ black square. For a grid to be tileable by dominoes, the number of white squares must equal the number of black squares.

Here: $8 \neq 6$

Since the number of white and black squares is unequal, the grid is not tileable.

Step 4 · Analyze the Conditional Statement

Let the propositions be:

$P$ : "The black-and-white grid is tileable"
$Q$ : "The plain grid is tileable"

Since both grids are the same shape, neither is tileable ( $P$ is false and $Q$ is false).

In logic, a conditional statement "If $P$ , then $Q$ " ( $P \implies Q$ ) is true whenever the hypothesis $P$ is false.

Answer

Yes, the statement is true.

Common Mistakes

Assuming Total Even Count Implies Tileability: Although $14$ is an even number, domino tiling requires equal counts of black and white squares ( $8 \neq 6$ ), which makes tiling impossible.
Misinterpreting Implication: For any implication $P \implies Q$ , if the premise $P$ is false, the implication is vacuously true regardless of the truth value of $Q$ .

Question 46

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