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Question:

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is

Explanation:

Let each square tile have side = 1 unit

Let the length of the rectangular floor be m units and the breadth be n units.

Number of red tiles = (m – 2)(n – 2)

Number of white tiles = mn − (m – 2)(n – 2)

Now, (m – 2)(n – 2) = mn − (m – 2)(n – 2)
⇒ 2(mn – 2m – 2n + 4) − mn = 0
⇒ mn – 4m – 4n + 8 = 0
⇒ n(m – 4) = 4m − 8 
⇒ n = 4(m – 2)/(m – 4)

 Now, consider the options.

Option (a): If m = 10, n = 32/6, which is not possible as n is an integer

Option (b): If m = 12, n = 40/8 = 5, which is possible

Option (c): If m = 14, n = 48/10, which is not possible as n is an integer

Option (d): If m = 16, n = 56/12, which is not possible as n is an integer

Hence, option (b).