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

Question

Homi · Accepted Answer

A region can only be tiled by dominoes if it has an equal number of black and white squares when colored like a checkerboard.

Step 1 — Count the squares in the region

Let us count the total number of squares in the region. The region is like a rectangle that is 3 squares wide. It is 5 squares high. A full 3x5 rectangle would have $3 \times 5$ squares.

$3 \times 5 = \textbf{15 squares}$

From the diagram, we see that one square is missing from the top row, in the middle. So, we subtract this 1 missing square from the total. The number of squares in the region is $15 - 1$ .

$15 - 1 = \textbf{14 squares}$

Diagram 1

Step 2 — Count black and white squares

Let us count the number of black and white squares in the region. We look at the "Black and white grid" diagram. We can count the white squares (W) and black squares (B) directly.

Row 1 has 2 white squares. Row 2 has 1 white square and 2 black squares. Row 3 has 2 white squares and 1 black square. Row 4 has 1 white square and 2 black squares. Row 5 has 2 white squares and 1 black square.

Let us add up the white squares.

$2 + 1 + 2 + 1 + 2 = \textbf{8 white squares}$

Let us add up the black squares.

$0 + 2 + 1 + 2 + 1 = \textbf{6 black squares}$

Diagram 2

Step 3 — Check tileability condition

Each tile is a domino. A domino covers exactly two squares. When placed on a checkerboard pattern, any domino will always cover one white square and one black square. For a region to be tileable by dominoes, it must have an equal number of white squares and black squares. In our region, we have 8 white squares. We have 6 black squares. The number of white squares is not equal to the number of black squares.

$8 \neq 6$

Since the number of white squares is not equal to the number of black squares, this region cannot be tiled by dominoes. This means the "black-and-white grid" is not tileable. It also means the "plain grid" is not tileable, because it is the same shape.

Step 4 — Answer the question based on logic

The question asks: "If the black-and-white grid is tileable, is the plain grid tileable?" Let us call the first part "P": "the black-and-white grid is tileable". Let us call the second part "Q": "the plain grid is tileable". We found that the black-and-white grid is not tileable. So, P is false. We also found that the plain grid is not tileable. So, Q is false. The question is asking if the statement "If P, then Q" is true. In logic, if the "if" part (P) is false, then the entire "if...then..." statement is considered true, regardless of whether Q is true or false. Since P is false, the statement "If P, then Q" is true.

Answer

Yes, the statement is true.

Question 46

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