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:

Trace which squares of the net become adjacent faces on the cube, and check whether the rotational order of the shaded pattern around the diamond corner is preserved without any mirror flip.

Answer 1

The Correct Option is A

Step 1: Pick one vertex of the cube that is shared by three faces visible in the net.
In the net, the diamond square touches the split triangle square along one shared edge, and that shared edge has two endpoints, each of which is a corner of the cube once folded. Focus on the corner where the diamond face, the split triangle face, and a third neighbouring face all come together.

Step 2: Write down the order of faces going around that corner in the flat net.
Standing at that shared corner and sweeping around it in the net, in a fixed direction (say clockwise as drawn on the page), the faces appear in the order: diamond face, then split triangle face, then the next face along the strip. This fixed clockwise order is a physical property of the paper and cannot change no matter how the paper is folded, since folding only bends the paper along creases, it never re-orders which faces meet at a corner or flips their sequence.

Step 3: Read off the same corner in cube (i) and check the order of faces.
Looking at cube (i) from the same viewing corner, the diamond face, the face carrying the larger part of the shaded triangle, and the adjoining side face with the thin shaded sliver appear in that same clockwise sequence as in the net, so cube (i) is consistent with a genuine fold.

Step 4: Read off the same corner in cube (ii) and check the order of faces.
In cube (ii), the position of the larger shaded triangle has moved to the opposite side face relative to the diamond, which reverses the sequence of faces around that corner compared to the net. A reversed sequence can only happen if the net were flipped over before folding, which is not a valid paper fold, so cube (ii) is inconsistent with Panel I.

Step 5: Conclude using the elimination of the impossible option.
Since checking the order of faces around a shared corner shows cube (i) matches the net's fixed sequence while cube (ii) requires an impossible mirror flip, only cube (i) can be produced by actually folding the paper shown in Panel I.

\[ \boxed{\text{Option (A): only (i) is achievable}} \]