Constructions and Tilings | IT

Question 45

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

Question diagram 1
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Solution

IT-45

Chapter: CONSTRUCTIONS AND TILINGS
Class: 7 (Class 7)
Category: in_text

Question

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

Question diagram(s):

Question diagram

A region can be tiled by dominoes only if it meets two important conditions. We will check these conditions for our region.

Step 1 — Count total squares

Let us count the small squares in the region. The region is a rectangle with 3 columns and 5 rows. A full 3×53 \times 5 rectangle would have 3×53 \times 5 squares. 3×5=153 \times 5 = 15 The diagram shows that two squares are removed from the top row. So, the total number of squares in the region is 15−215 - 2. 15−2=1315 - 2 = \mathbf{13} Each tile covers 2 squares. For a region to be tiled by these tiles, the total number of squares must be an even number. Since 13 is an odd number, the plain grid cannot be tiled.

Diagram 1

Step 2 — Count black and white squares

Let us look at the black-and-white grid. We count the number of black and white squares in the region. The full 3×53 \times 5 grid has 15 squares. Let us assume the top-left square is white, as shown in the diagram. We can list the colors for each row: Row 1: White, Black, White Row 2: Black, White, Black Row 3: White, Black, White Row 4: Black, White, Black Row 5: White, Black, White In the full grid, there are 8 white squares and 7 black squares. The two removed squares are the top-left and top-right squares. Both of these removed squares are white. So, we remove 2 white squares from the count. Number of white squares remaining: 8−2=68 - 2 = \mathbf{6} Number of black squares remaining: 7\mathbf{7} For a region to be tiled by dominoes on a black-and-white grid, it must have an equal number of black and white squares. Here, we have 6 white squares and 7 black squares. They are not equal. So, the black-and-white grid cannot be tiled.

Diagram 2

Answer

The question asks: "If the plain grid is tileable, is the black-and-white-grid tileable?" From Step 1, we found that the plain grid is not tileable. This is because it has 13 squares, which is an odd number. Since the plain grid is not tileable, the "if" condition in the question is not met. Therefore, we cannot assume the plain grid is tileable. Based on our analysis in Step 2, the black-and-white grid is also not tileable. This is because it has an unequal number of black and white squares (6 white and 7 black).

The answer is No.

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Q28

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Q36

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Q38

Is an m×nm \times n grid tileable with 2×12 \times 1 tiles, if both mm and nn are even? If yes, come up with a general strategy to tile it.

Q39

Is an m×nm \times n grid tileable with 2×12 \times 1 tiles, if one of mm and nn is even and the other is odd? If yes, come up with a general strategy to tile it.

Q40

Is an m×nm \times n grid tileable with 2×12 \times 1 tiles, if both mm and nn are odd? Give reasons.

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

Is the following region tileable with 2×12 \times 1 tiles?

Q43

Context: We are considering whether a given region can be tiled using 2×12 \times 1 tiles.

Q. What about this one?

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 5×35 \times 3 grid, makes it non-tileable?

Q45

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 × 3 grid, makes it non-tileable?

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