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

Question

Homi · Accepted Answer

A grid can be tiled if its total area is covered by tiles.

Step 1 — Total squares check

Let $m$ be the number of rows. Let $n$ be the number of columns. The grid has $m \times n$ squares in total. We are told $m$ is an even number. We are told $n$ is an even number. So, $m$ can be written as $2 \times k$ . Here $k$ is a whole number. Also, $n$ can be written as $2 \times j$ . Here $j$ is a whole number. Let us find the total number of squares.

Total squares = $m \times n$ $= (2 \times k) \times (2 \times j)$ $= 4 \times k \times j$ This number is a multiple of 4. Any multiple of 4 is also a multiple of 2. Each $2 \times 1$ tile covers 2 squares. So, the total number of squares must be even. Our grid has an even number of squares. This means it might be possible to tile it.

$\boxed{\text{The total number of squares is even.}}$

Diagram 1

Step 2 — Tiling strategy

We know $m$ and $n$ are both even. This means we can divide $m$ by 2. This means we can divide $n$ by 2. Let us imagine dividing the grid. We can divide it into many small $2 \times 2$ squares. There will be $m/2$ rows of these $2 \times 2$ squares. There will be $n/2$ columns of these $2 \times 2$ squares. Let us look at just one $2 \times 2$ square. It has 4 small squares inside it. We need to cover these 4 squares. We can use two $2 \times 1$ tiles. Let us place the first $2 \times 1$ tile horizontally. It covers two squares in the top row. Let us place the second $2 \times 1$ tile horizontally. It covers two squares in the bottom row. This completely covers the $2 \times 2$ square. We can do this for every $2 \times 2$ square. This strategy will tile the entire grid.

$\boxed{\text{Yes, the grid is tileable.}}$

Diagram 2

Answer

(i) Yes, an $m \times n$ grid is tileable with $2 \times 1$ tiles if both $m$ and $n$ are even. (ii) A general strategy is to divide the $m \times n$ grid into $2 \times 2$ squares. Then, tile each $2 \times 2$ square with two $2 \times 1$ tiles placed horizontally.

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