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?
Show Hint
Label each square of the net as a face of the cube, remember opposite faces never appear together in one view, then trace where the shaded corner lands after folding.
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.
Show Solution
The Correct Option isB
Solution and Explanation
Step 1: Give each square in the net a number. Number the six squares of Panel I as 1 to 6, following the plus shape, square 1 is the topmost square, square 2 is the one below it in the middle of the row, square 3 and square 4 are the squares to the left and right of square 2, square 5 is below square 2, and square 6 is the last square at the bottom. This numbering just names the faces, it does not assume which one is the front yet.
Step 2: Fold the net one square at a time like rolling a die. Start with square 2 lying flat as the base. Fold square 1 up and back over square 2, it now stands as the face directly above square 2. Fold squares 3 and 4 up on either side of square 2, they become the two faces to the left and right of square 2. Fold square 5 down under square 2 and square 6 folds again under square 5, wrapping all the way to the far side of the cube, opposite square 2. After this rolling, square 2 and square 6 sit on opposite faces of the cube, square 1 and square 5 sit on opposite faces, and square 3 and square 4 sit on opposite faces.
Step 3: Place the shading on the rolled up cube. Squares 3 and 4 are the fully dark squares in the net, so two opposite faces of the finished cube are fully dark. Square 1 carries the corner triangle and square 6 carries the patch of shading, while squares 2 and 5 stay plain. Because square 1 is joined to square 2 along one particular edge, the shaded triangle on square 1 always ends up on the corner of the cube that is farthest from that shared edge once the fold is done, never on the corner nearest square 2.
Step 4: Test cube (i) against this rolled up shape. Cube (i) draws the shaded triangle right up against the corner where the top meets the visible side face closest to the viewer. That is the corner nearest the fold with square 2, which Step 3 rules out, so cube (i) does not match the net.
Step 5: Test cube (ii) against this rolled up shape. Cube (ii) places the shaded triangle on the far corner of the top face, away from the front edge, and its visible side face is fully dark to match squares 3 or 4. Rolling the net physically produces exactly this arrangement, so cube (ii) is a valid fold of the net.
Final Answer: Rolling the die shows that only cube (ii) can be the folded form of the net in Panel I, which is option (B). \[ \boxed{\text{Only (ii)}} \]
