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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Fold the net without flipping it over: the order in which shaded faces meet at a shared corner in the flat net must stay the same after folding, never mirrored.
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 isB
Solution and Explanation
Step 1: Understanding the Question.
Panel I shows a flat six-square paper cutout (a cube net) with certain regions shaded. When this net is folded along the dashed lines, it forms a cube, and the shaded regions land on specific outer faces. We must check which of the two isometric cubes in Panel II, (i) or (ii), can actually be produced by folding the net exactly as drawn, without flipping the sheet over.
Step 2: Key Formula or Approach.
Two rules make this kind of question solvable without physically cutting and folding paper. First, in a net, a face and the face directly two squares away in a straight line become opposite faces of the cube once folded, while directly touching squares become adjacent faces. Second, and most useful here, folding a flat sheet is a rigid operation: it never mirrors or flips the sheet over. So the rotational order (clockwise or anticlockwise) in which markings sit around any shared corner or edge in the flat net must stay the same after folding. If a candidate cube shows the same markings in the opposite rotational order, that cube is a mirror image and cannot be produced by folding this particular net.
Step 3: Detailed Explanation.
Pick the dashed central square in Panel I as the reference (front) face; the squares directly above and below it fold to become the top and bottom faces, and the two squares attached on its left and right fold to become the left and right faces, while the square at the far end of the vertical strip becomes the back face, opposite the front.
Two shaded regions are the most distinctive markings on the net: the gray L-shaped patch with a white notch on the left-arm square, and the gray diagonal triangle on the topmost square. Once folded, these two shaded faces sit next to each other around a shared corner of the cube, and the straight edge of the L-shaped patch meets the diagonal cut of the triangular patch at a fixed rotational sense, set entirely by how they are drawn touching each other in the flat net.
Checking cube (i): the diagonal shaded face and the L-shaped shaded face meet at their shared edge in the reversed rotational sense compared to the net, which only happens if the net were flipped over before folding. Since we are only folding, never flipping, cube (i) cannot be the correct result.
Checking cube (ii): the shaded faces meet at their shared edge in the same rotational sense as they appear in the flat net in Panel I, exactly what a genuine fold, with no flip, produces. So cube (ii) is a valid, achievable folding of the net.
Step 4: Final Answer.
Only cube (ii) can be obtained by folding the paper shown in Panel I, so the correct option is (B).
\[ \boxed{\text{Only (ii)}} \]
