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?
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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).
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Approach Solution -2
Since the final figure is simply the union of the black squares from rotated P and rotated Q, counting how many squares are black in total for each option is a fast way to check which one can be right, because the union can never have more black squares than the two rotated grids combined and never fewer than the larger of the two.
Option A: shows a red square and six black squares out of the remaining eight cells, which matches the total produced once the two rotated grids are combined and their overlaps merged.
Option B: shows too few black squares to account for every position that is black in either rotated grid, so some squares that should stay black after the union are missing.
Option C: places black squares in positions that neither rotated grid actually occupies, which cannot happen since a union can only add squares that were already black in at least one of the two grids.
Option D: gets the count of black squares right but places the red square in the wrong cell, which does not match where both rotated grids agree the red square should sit.
Only Option A has the black squares and the red square landing exactly where the union of the two rotated grids requires.
