P and Q are two grids, in which white squares are transparent. P is rotated 90 degrees counter-clockwise and Q is rotated 90 degrees clockwise. What would be the resulting figure if the rotated grids are overlapped?
Show Hint
When dealing with grid rotations, it can be helpful to sketch the results on paper. To rotate a point (x,y) 90 degrees counter-clockwise around the center of a 3x3 grid, track how the corners and center move. For overlapping, remember that transparent + colored = colored, and colored + colored = colored.
Step 1: Understanding the Concept:
This problem involves two spatial transformations (rotation) followed by a superposition (overlapping). We must perform each step accurately to find the final result.
Step 2: Detailed Explanation:
Rotate Grid P:
Grid P must be rotated 90 degrees counter-clockwise.
The red square at the top-left corner will move to the bottom-left corner.
The column of three black squares will become a horizontal row at the bottom.
The other black squares will rotate accordingly. The resulting rotated P will have black squares at (row 3, col 1-Red), (row 3, col 2), (row 3, col 3), (row 1, col 2), (row 1, col 3), (row 2, col 3).
Rotate Grid Q:
Grid Q must be rotated 90 degrees clockwise.
The red square at the bottom-right corner will move to the bottom-left corner.
The checkerboard pattern will rotate. The resulting rotated Q will have black squares at (row 3, col 1-Red), (row 1, col 1), (row 3, col 1), (row 2, col 2), (row 1, col 3), (row 3, col 3).
Overlap the Rotated Grids:
Now we combine the two rotated grids. A square in the final grid is colored if it is colored in either of the rotated grids.
Red Square: In rotated P, the bottom-left square is red. In rotated Q, the bottom-left square is also red. When overlapped, the bottom-left square will be red.
Black Squares: We take the union of all the black square positions from both rotated grids.
From Rotated P: (3,2), (3,3), (1,2), (1,3), (2,3).
From Rotated Q: (1,1), (2,2), (1,3), (3,3).
Combined Black Positions: (1,1), (1,2), (1,3), (2,2), (2,3), (3,2), (3,3).
Construct the Final Grid:
The final grid has a red square at (3,1) and black squares at all other positions except (2,1) and (3,1), which is empty transparent, and (3,1) which is red. Actually, the square (3,1) is red, and the square (2,1) is transparent. The top row is all black. The middle row is transparent, black, black. The bottom row is red, black, black. This matches the figure in option A.
Step 3: Final Answer:
After performing the rotations and overlapping the grids, the resulting figure is the one shown in option (A).
