A 3 x 3 grid is colored using red and blue colors such that if we rotate the…
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A 3 x 3 grid is colored using red and blue colors such that if we rotate the grid about its center in the plane by 180 degrees, the grid looks the same. The number of ways are
- A.
72
- B.
27
- C.
32
- D.
96
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Correct answer: C
Answer: 32
Reasoning:
Label the 3×3 grid positions as: 1 2 3 / 4 5 6 / 7 8 9
Under a 180° rotation the swapped pairs are: 1 ↔ 9, 2 ↔ 8, 3 ↔ 7, 4 ↔ 6. The center cell 5 maps to itself.
Each swapped pair must be the same color, so each pair gives 2 choices (red or blue). The center cell also has 2 choices.
There are 4 independent pairs plus the center = 5 independent positions, so the total number of valid colorings is 2^5 = 32.
Therefore, the number of colorings fixed by a 180° rotation is 32.