Is an grid tileable with tiles, if both and are even? If yes, come up with a general strategy to tile it.

Constructions and Tilings | IT

Question 38

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.

Solution

Understand the Question

A single $2 \times 1$ tile covers $2$ unit squares.
An $m \times n$ grid has a total of $m \times n$ unit squares.
If both $m$ and $n$ are even, the grid can be decomposed into smaller $2 \times 2$ squares, each of which can be tiled by two $2 \times 1$ tiles.

Step 1 · Check Total Number of Squares

Let $m = 2k$ and $n = 2j$ , where $k$ and $j$ are positive integers since $m$ and $n$ are both even. Diagram 1

\begin{aligned} \text{Total squares} &= m \times n \\ &= (2 \times k) \times (2 \times j) \\ &= 4 \times k \times j \end{aligned}

Since the total number of squares is a multiple of $4$ (and therefore divisible by $2$ ), the total area can potentially be covered by $2 \times 1$ tiles.

Step 2 · Formulate Tiling Strategy

Since $m$ and $n$ are both even, we can partition the $m \times n$ grid into $\dfrac{m}{2} \times \dfrac{n}{2}$ smaller blocks of size $2 \times 2$ . Diagram 2

Each $2 \times 2$ square consists of $4$ unit squares and can be completely covered by placing two $2 \times 1$ tiles horizontally (or vertically).

Repeating this placement for all $\dfrac{m}{2} \times \dfrac{n}{2}$ blocks tiles the entire $m \times n$ grid completely without any gaps or overlaps.

Answer

Yes, an $m \times n$ grid is tileable.

Strategy: Divide the $m \times n$ grid into $\dfrac{m}{2} \times \dfrac{n}{2}$ blocks of size $2 \times 2$ , and cover each $2 \times 2$ block with two $2 \times 1$ tiles placed horizontally.

Common Mistakes

Assuming Area Divisibility is Enough: While having an even total area is necessary to tile with $2 \times 1$ tiles, a valid tiling requires a concrete placement strategy without gaps or overlaps.
Overcomplicating the Pattern: Since both dimensions are even, simply subdividing into $2 \times 2$ blocks (or filling row-by-row) is sufficient rather than trying alternating or complex checkerboard patterns.

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Q28

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Construct a regular hexagon of sidelength $5\text{ cm}$ .

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Can a $4 \times 6$ grid be tiled using multiple copies of $2 \times 1$ tiles? We are allowed to rotate a $2 \times 1$ tile and use it.

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Can a $4 \times 7$ grid be tiled using $2 \times 1$ tiles?

Q36

What about a $5 \times 7$ grid?

Complete the justification.

Q38

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.

Q40

Is an $m \times n$ grid tileable with $2 \times 1$ tiles, if both $m$ and $n$ are odd? Give reasons.

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?

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

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?