Discussion
The figure shows two 4-tile patterns.
Either one or both of the patterns can be used any number of times and in any orientation to construct a new pattern. Which one of the options below cannot be constructed by using only these two 4-tile patterns assuming there are no overlaps among them?
- A.
- B.
- C.
- D.
Answer: C
The pattern that cannot be built is (C), the 3 by 5 rectangle.
Both building blocks contain exactly 4 tiles, and pieces may not overlap, so any pattern made only from them must contain a multiple of 4 tiles. Count the tiles in each option:
- (A) 2 × 4 = 8 tiles: two 2 × 2 squares side by side, or two lines of 4 stacked.
- (B) 3 × 4 = 12 tiles: three horizontal lines of 4.
- (C) 3 × 5 = 15 tiles: not a multiple of 4.
- (D) 4 × 5 = 20 tiles: five vertical lines of 4.
Since 15 is not divisible by 4, no combination of the two patterns can cover the 3 by 5 rectangle exactly, while each of the others is built as shown.
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