Question
A $4m \times 4m$ floor needs to be covered by tiles of size $2m \times 1m$. Two diagonally opposite corners of size $1m \times 1m$ should be left uncovered. How many tiles are required to complete the job without breaking the tiles or overlapping them?
The correct answer is
Impossible to cover
The problem involves covering a 4m x 4m floor with 2m x 1m tiles while leaving two diagonally opposite corners of size 1m x 1m uncovered.
Let's solve this step by step:
- Calculate the total area of the floor: 4 \times 4 = 16 \, \text{m}^2.
- Calculate the area that needs to be left uncovered: 1 \times 1 + 1 \times 1 = 2 \, \text{m}^2.
- Thus, the area to be covered by tiles is: 16 - 2 = 14 \, \text{m}^2.
- Each tile covers: 2 \times 1 = 2 \, \text{m}^2.
- The number of tiles required theoretically is: \frac{14}{2} = 7.
- However, placing 7 tiles each covering 2m2 is impossible without cutting or overlapping them while leaving two 1m2 corners uncovered, due to the arrangement and size constraints.
- A mathematical rule regarding tiling problems states that if two opposite corners of a checkerboard are removed, it is impossible to cover the remaining area with dominos that cover two adjacent squares (which is similar to our tiles).
Therefore, the correct answer is that it is Impossible to cover the floor with the given conditions.
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