A paper shown in Panel I is folded along the dashed lines (- - -) to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube. Referring to cubes shown in Panel II, which one of the options is correct?
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Mentally fold the net along the dashed lines, track which two shaded faces end up sharing an edge, and check whether their relative orientation matches cube (i) or cube (ii).
Only (i) can correspond to the unfolded cube in Panel I.
Only (ii) can correspond to the unfolded cube in Panel I.
Both (i) and (ii) can correspond to the unfolded cube in Panel I.
Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.
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The Correct Option isA
Solution and Explanation
Step 1: Understanding the Question.
Panel I shows a flat net of six squares joined by dashed fold lines, with shaded shapes drawn on some of the squares. When the net is folded along every dashed line, it forms a closed cube, and the shaded shapes end up sitting on the outer faces of that cube. Panel II shows two finished cubes, (i) and (ii), each displaying a shaded diamond on one face and a shaded triangle on a face next to it. We must decide which of these two cubes could actually result from folding the net in Panel I.
Step 2: Key Approach.
For this type of net-folding question, pick two shaded faces in the net that share a common fold edge, since folding brings them together while keeping their edges glued along that shared line. Once folded, the corner of one shaded shape that touches the shared fold edge must line up with the matching corner of the neighboring shaded shape, in a fixed relative rotation. Any cube that shows these same two shapes with a different relative rotation is not a valid fold of the net, and must be rejected.
Step 3: Detailed Explanation.
In the net, the square carrying the gray diamond sits directly next to the square carrying a shaded triangle, joined by a dashed fold line. Folding along that line brings these two squares together edge to edge on the finished cube, so on the real cube the diamond face and the triangle face must be adjacent faces, and the triangle's shaded corner must point toward the diamond in one fixed way, fixed by how the paper was drawn before folding.
Checking cube (i) in Panel II: the diamond face and the triangle face sit next to each other with the triangle's shaded corner pointing toward the diamond in the same way the flat net dictates. This matches the fold exactly.
Checking cube (ii): the triangle on the adjacent face is rotated into a different corner compared to what the net produces when folded correctly. This mismatch means cube (ii) cannot be obtained by folding the given net, no matter how it is rotated in space, since rotating a solid cube in your hand never changes which corners of two adjacent shapes touch each other.
Step 4: Final Answer.
Only cube (i) has its diamond and triangle faces in the correct relative orientation demanded by the fold lines in Panel I, so cube (ii) must be rejected.
\[ \boxed{\text{Only (i) can correspond to the unfolded cube in Panel I.}} \]
