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
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
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Let the floor be m tiles by n tiles. Red tiles form an (m-2) by (n-2) block. Set white = red and factor.
Updated On: Nov 4, 2026
- 10
- 12
- 14
- 16
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The Correct Option is B
Solution and Explanation
Step 1: Set up:
Let the floor have \(m\) tiles along one edge and \(n\) along the other. The interior (red) tiles form an \((m-2) \times (n-2)\) block, so red \(= (m-2)(n-2)\) and white \(= mn - (m-2)(n-2) = 2m + 2n - 4\).
Step 2: Equate white and red:
\((m-2)(n-2) = 2m + 2n - 4 \Rightarrow mn - 2m - 2n + 4 = 2m + 2n - 4 \Rightarrow mn - 4m - 4n + 8 = 0 \Rightarrow (m - 4)(n - 4) = 8\).
Step 3: Solve in positive integers:
Both \(m, n\) must exceed 2. A negative factor, say \(m - 4 = -1\), would force \(n - 4 = -8\), i.e. \(n < 0\), which is impossible. So the factors are positive. Factor pairs of 8: \((1, 8), (2, 4), (4, 2), (8, 1)\), giving floors \(5 \times 12\) and \(6 \times 8\). Check: \(5 \times 12\): 60 tiles, 30 red, 30 white. \(6 \times 8\): 48 tiles, 24 red, 24 white.
Step 4: Choose the option:
Possible edge lengths are 5, 12, 6 and 8. Of the options 10, 12, 14 and 16, only \(12\) appears.
Final Answer:
The correct option is (2): 12
Let the floor have \(m\) tiles along one edge and \(n\) along the other. The interior (red) tiles form an \((m-2) \times (n-2)\) block, so red \(= (m-2)(n-2)\) and white \(= mn - (m-2)(n-2) = 2m + 2n - 4\).
Step 2: Equate white and red:
\((m-2)(n-2) = 2m + 2n - 4 \Rightarrow mn - 2m - 2n + 4 = 2m + 2n - 4 \Rightarrow mn - 4m - 4n + 8 = 0 \Rightarrow (m - 4)(n - 4) = 8\).
Step 3: Solve in positive integers:
Both \(m, n\) must exceed 2. A negative factor, say \(m - 4 = -1\), would force \(n - 4 = -8\), i.e. \(n < 0\), which is impossible. So the factors are positive. Factor pairs of 8: \((1, 8), (2, 4), (4, 2), (8, 1)\), giving floors \(5 \times 12\) and \(6 \times 8\). Check: \(5 \times 12\): 60 tiles, 30 red, 30 white. \(6 \times 8\): 48 tiles, 24 red, 24 white.
Step 4: Choose the option:
Possible edge lengths are 5, 12, 6 and 8. Of the options 10, 12, 14 and 16, only \(12\) appears.
Final Answer:
The correct option is (2): 12
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