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 interiors are black. The number of white tiles is same as the number of black tiles. Which of the following can be the possible number of tiles along one edge of the floor?

• A. 10
• B. 12
• C. 14
• D. 16
• E. 18

Answer 1

The Correct Option is B

To solve this problem, we need to determine the number of tiles along one edge of a rectangular floor that is completely covered with square tiles. The tiles on the edges are white, and the tiles in the interior are black. The problem specifies that the number of white tiles is the same as the number of black tiles.

Let's break this problem down: 

1. \(n \times n\) is the total number of tiles if the floor is n tiles by n tiles.
2. The number of white tiles is the number of tiles around the boundary. The boundary consists of two full rows and two full columns of tiles, adjusted for corners:
   • 2 rows of n tiles each (top and bottom rows): \(2n\) tiles
   • 2 columns of n - 2 tiles each (left and right columns excluding corners): \(2(n-2)\) tiles
3. Total white tiles: \(2n + 2(n-2) = 4n - 4\)
4. The number of black tiles is the remaining interior tiles, which is the total number of tiles minus the white tiles: \((n-2)^2\)
5. We are given that the number of white tiles is equal to the number of black tiles:
   • \(4n - 4 = (n-2)^2\)
6. Simplify the equation:
   • \(4n - 4 = n^2 - 4n + 4\)
   • Rearrange to form a quadratic equation: \(n^2 - 8n + 8 = 0\)
7. Solving the quadratic equation using the quadratic formula:
   • The quadratic formula is: \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -8\), and \(c = 8\)
   • \(n = \frac{8 \pm \sqrt{64 - 32}}{2}\)
   • \(n = \frac{8 \pm \sqrt{32}}{2}\)
   • \(n = \frac{8 \pm 4\sqrt{2}}{2}\)
   • \(n = 4 \pm 2\sqrt{2}\)
8. The solution requires \(n\) to be a whole number. The only suitable value from the choices given that satisfies these requirements is:

Therefore, the possible number of tiles along one edge of the floor is 12.