Question

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?

• A. Only (i) can correspond to the unfolded cube in Panel I.
• B. Only (ii) can correspond to the unfolded cube in Panel I.
• C. Both (i) and (ii) can correspond to the unfolded cube in Panel I.
• D. Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.

Hint:

Find the one face in the net that is a single solid block of gray with no white in it, then check which cube shows that same solid gray face together with its correctly matched neighbors.

Answer 1

The Correct Option is B

Step 1: Label every face of the net.
The net in Panel I is a cross-shaped arrangement of six squares. Call the hub square (the one with fold lines on all four sides) Face C. Above it is Face T, below it is Face D1, and below D1 is Face D2. To the left of C is Face L, and to the right of C is Face R.

Step 2: Work out which faces end up opposite each other.
When a net folds into a cube, two faces separated by exactly one square in a straight line become opposite faces, because the paper wraps all the way around. Going down the vertical strip T-C-D1-D2, this gives two opposite pairs: T is opposite D1, and C is opposite D2. The two side squares attached to the hub give the third pair: L is opposite R.

Step 3: Read off the shading on each face.
Face D2, at the very bottom of the net, is fully shaded gray with no white area at all, so it is a solid gray face. Face C, the hub, is completely unshaded (white). Face T has a small gray triangular patch tucked into the corner nearest the fold that joins it to C. Face D1 is unshaded (white). Face L is mostly gray with a small white square notch cut out of one corner. Face R is mostly white with a small gray square patch in one corner.

Step 4: Use the fully-gray face as an anchor.
Face D2 is the only face on the whole net that is a single solid block of gray with no white part in it. This is a strong, easy-to-spot fingerprint: any cube honestly folded from this net must show one, and only one, completely solid gray face, and the face directly opposite it (Face C) must be completely white with no shading.

Step 5: Check cube (i) against this fingerprint.
In cube (i), none of the visible faces is a single solid block of gray. The large face is gray with a white notch cut into it (this looks like Face L's pattern), the top face carries only a thin gray sliver rather than a full solid fill, and the narrow side face is split between white and gray. Because Face D2's solid-gray fingerprint does not show up anywhere, and the visible faces are combined in an orientation that a real fold of this net cannot produce, cube (i) cannot be built from this net.

Step 6: Check cube (ii) against this fingerprint.
In cube (ii), one full face (the top) is a solid, unbroken block of gray, exactly matching Face D2. The face that would sit opposite it is plain white, matching Face C. The remaining visible face carries a single diagonal gray triangle in one corner, which is consistent with the small corner patches carried by Face T or Face R once the net is folded and rotated so that D2 faces up. Every visible face in cube (ii) matches a face of the net in both shape and relative position.

Final Answer:
Cube (ii) is consistent with folding the given net and cube (i) is not, so only (ii) can correspond to the unfolded cube in Panel I. \[ \boxed{\text{Option (B)}} \]