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.
Instead of tracking where the black squares end up, it is quicker here to track where the white, transparent squares end up, since a cell only stays transparent in the final figure if it is white in both rotated grids.
Rotating P by 90 degrees counter-clockwise moves its transparent squares to a new set of positions, and rotating Q by 90 degrees clockwise moves its transparent squares to another set.
Comparing the two rotated sets, only one cell, apart from the red square's cell, stays transparent in both grids at once, so that is the only cell that should remain white in the final overlap.
Option A is the only option that leaves exactly that one cell transparent while colouring every other cell black or red, so it matches the required overlap.
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Approach Solution -2
A third approach is to track only the red square, since it is the single easiest feature to follow through both rotations, and then use its final position as an anchor to judge the rest of each option.
Option A: places the red square at the bottom-left cell, which is exactly where it lands after rotating its starting corner position 90 degrees counter-clockwise in P and 90 degrees clockwise in Q, since both rotations send it to the same corner.
Option B: places the red square correctly but gets the surrounding black pattern wrong, so it fails once the rest of the grid is checked against the anchor.
Option C: puts the red square in a different cell altogether, which cannot be right since both rotations agree on where it should end up.
Option D: also shifts the red square away from the corner both rotations agree on, ruling it out immediately.
Since Option A is the only option that anchors the red square correctly and keeps the rest of the pattern consistent with it, it is the correct resulting figure.
