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
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.
Show Solution
The Correct Option isB
Solution and Explanation
Step 1: Understanding the Question.
We are given a flat net of 6 squares in Panel I, with some squares shaded, and told it folds into a cube along the dashed lines. We must decide which of the two drawn cubes in Panel II, (i) or (ii), matches the actual folded result.
Step 2: Key Rule or Approach.
Use the opposite-face rule for a net that is a straight strip of four squares with two extra squares attached as side flaps on one of the middle squares of the strip: the 1st and 3rd squares of the four-square strip become opposite faces of the cube, the 2nd and 4th squares become opposite faces, and the two side flaps become the remaining opposite pair. Once every face is identified by this rule, an isometric cube drawing can only be valid if the three faces it shows meeting at one visible corner are faces that are genuinely pairwise adjacent, never opposite, in the net, and if the shading on each visible face keeps the same edge that touched its shaded neighbor in the flat net.
Step 3: Detailed Explanation.
Number the vertical strip of four squares from top to bottom as 1, 2, 3, 4, where square 2 is the dashed reference square with the two side flaps (call them 5 and 6) attached to it. By the opposite-face rule: square 1 is opposite square 3, square 2 is opposite square 4, and squares 5 and 6 are opposite each other. Square 2, along with squares 1, 5, and 6, are therefore the four faces that surround square 2 on all sides, so any three of these four can appear together at a corner of the cube in an isometric view, but square 4, opposite square 2, never appears in the same view as square 2.
Now look at option (i): the visible corner in this drawing places the two shaded faces from squares 1 and 5, the diagonal patch and the L-shaped patch, next to each other in a way that would require square 5's shaded edge to sit against a different edge of square 1 than the one it actually touches in the flat net. That contradicts the fixed adjacency set by the net, so option (i) is not achievable.
Option (ii), however, places these same two shaded faces together at the corner using exactly the edges that are joined in the flat net, so it is consistent with a genuine fold.
Step 4: Final Answer.
Working through the opposite-face pairing and checking which candidate keeps the correct shared edges confirms that only cube (ii) is a valid folding of the net in Panel I, so the answer is option (B).
\[ \boxed{\text{Only (ii)}} \]
