Constructions and Tilings | IT

Question 41

Here is a 5×35 \times 3 grid, with a unit square removed. Now, it has an even number of unit squares. Is it tileable with 2×12 \times 1 tiles?

Question diagram 1
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Solution

We will use a chessboard coloring method to solve this problem.

Step 1 — Count the squares

First, let us find the total number of unit squares in the grid. The grid has 5 rows. It has 3 columns. So, the total number of squares is:

5×35 \times 3

=15= 15

One unit square is removed from this grid. Let us find the number of squares remaining.

15−115 - 1

14 squares\boxed{14 \text{ squares}}

Each 2×12 \times 1 tile covers exactly two unit squares. So, if the grid can be tiled, it must have an even number of squares. Our grid has 14 squares, which is an even number. This means tiling might be possible.

Diagram 1

Step 2 — Use chessboard coloring

Let us color the grid like a chessboard. We can start with a black square at the top-left corner. Let's count the number of black and white squares.

Original 5×35 \times 3 grid coloring (B = Black, W = White): B W B W B W B W B W B W B W B

Let us count the black squares. Row 1 has 2 black squares. Row 2 has 1 black square. Row 3 has 2 black squares. Row 4 has 1 black square. Row 5 has 2 black squares. Total black squares:

2+1+2+1+22 + 1 + 2 + 1 + 2

=8= 8

Let us count the white squares. Row 1 has 1 white square. Row 2 has 2 white squares. Row 3 has 1 white square. Row 4 has 2 white squares. Row 5 has 1 white square. Total white squares:

1+2+1+2+11 + 2 + 1 + 2 + 1

=7= 7

So, the original grid has 8 black squares and 7 white squares. The removed square is the center square. Looking at our coloring, the center square (row 3, column 2) is White. When this white square is removed, the number of white squares changes. The number of black squares remains the same.

Remaining black squares: 8 Remaining white squares:

7−17 - 1

=6= 6

For a grid to be tileable by 2×12 \times 1 tiles, it must have an equal number of black and white squares. Our remaining grid has 8 black squares and 6 white squares. These numbers are not equal. So, the grid cannot be tiled by 2×12 \times 1 tiles.

Answer

No, the grid is not tileable with 2×12 \times 1 tiles.

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Q41

Here is a 5×35 \times 3 grid, with a unit square removed. Now, it has an even number of unit squares. Is it tileable with 2×12 \times 1 tiles?

Q42

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Q43

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