NCERT Solutions for Class 7th Maths Chapter 6 Are the following tilings possible? — Figure it Out
Book page 160 Updated on2026-09-19
Q1.
Are the following tilings possible? 1. Region to be tiled: a 12-square staircase region (two rows of 2 squares on top of two rows of 4 squares). Tile: an L-shaped tile made of 3 squares.
Answer
Yes, it is possible. Four L-shaped tiles cover the region exactly.
Squares in the region = (2 rows × 2) + (2 rows × 4)
= 4 + 8 = 12
Each L-tile covers 3 squares
Number of tiles = 12 ÷ 3 = 4
One possible tiling. Blue and orange interlock on the left; green and violet finish the bottom-right.
The four tiles, written out. Number the rows 1 to 4 from the top and the columns 1 to 4 from the left.
Tile
Squares it covers
Shape
Blue
(1,1), (1,2), (2,1)
L
Green
(2,2), (3,2), (3,3)
L
Orange
(3,1), (4,1), (4,2)
L
Violet
(3,4), (4,4), (4,3)
L
Why it happens: 12 is a multiple of 3, so the counting test is passed. But passing it is not enough — an arrangement must actually be found, and here one does exist. The trick is the corner square (4,1): it can only be reached by a tile that also uses (3,1) and (4,2), and once that tile is fixed the rest follows.
Check it yourself: all 12 squares appear exactly once in the table above — count them.
Q2.
Are the following tilings possible? 2. Region to be tiled: an 8 × 8 grid with two opposite corner squares removed. Tile: a 2 × 1 tile.
Answer
No — this one is impossible, even though the count of squares is even.
8 × 8 = 64 squares; remove 2 corners ⇒ 62 squares
62 is even, so 31 tiles would be needed — the counting test is passed.
Now colour like a chessboard. In a full 8 × 8 board:
32 light squares and 32 dark squares
The two removed corners are diagonally opposite,
and diagonally opposite corners are always the same colour.
Left after removal: 32 light and 30 dark
Each tile takes one light and one dark ⇒ 31 tiles need 31 of each
32 ≠ 31 ⇒ tiling is impossible
The two removed corners (dashed) are both dark. 32 light squares are left against only 30 dark ones.
Why it happens: a domino must always straddle two neighbouring squares, and neighbouring squares are always of opposite colours. So the 31 dominoes would need 31 light and 31 dark squares. Cutting two squares of the same colour breaks that balance permanently, and no cleverness in arranging the tiles can repair it.
Did you know? This is the famous mutilated chessboard problem. It is the same colouring argument used for Fig. 6.13, on a bigger board.