The sheet of paper shown in the figure (X) given on the left hand side, in…

When a strip of four faces in a net is folded into a box, the 1st and 3rd faces in the strip fold to become opposite faces, and the 2nd and 4th faces fold to become opposite faces; a flap attached above one of these faces and a flap attached below the same face fold to become the box's top and bottom faces, which are opposite each other regardless of which face in the strip they are attached to. Consequently, in any three-face corner view of the folded box, the three visible faces must all be mutually adjacent - a genuine view can never place two faces that are actually opposite each other side by side.

In sheet (X), the horizontal strip has four faces in order: the first face is plain (white), the second face is plain (white) and carries a shaded flap above it and a shaded flap below it, the third face is plain (white), and the fourth face is shaded (black). Applying the rule above: the first and third faces are opposite each other (both plain), the second and fourth faces are opposite each other (one plain, one shaded), and the top and bottom flaps are opposite each other (both shaded). So the box has three shaded faces in total - the fourth strip face and the two flaps - but only the two flaps are actually opposite one another; the shaded fourth face is adjacent to both flaps and to the two plain end faces.

Checking each figure against this rule:

• Figure (1): one shaded face on top together with one shaded side face and one plain side face - the shaded top flap and the shaded fourth face genuinely are adjacent, so this view is achievable.

• Figure (2): the two side faces shown are both shaded while the top is plain - but only one strip face is ever shaded, so two shaded side faces can never be adjacent to each other; this view cannot occur.

• Figure (3): one shaded side face together with a plain top and a plain second side - again matches the single shaded strip face sitting next to plain faces; achievable.

• Figure (4): all three visible faces are shaded - that would need both shaded flaps (which are opposite each other and never both show up in the same corner view) to appear together with the shaded strip face; this view cannot occur.

As an independent cross-check on this conclusion: the only two faces that could ever combine with the shaded fourth face to make all three visible faces shaded are the top and bottom flaps - but the top and bottom flaps are themselves opposite faces of the box, so a genuine corner view can show at most one of them, never both together with the shaded fourth face. A fully-shaded three-face view is therefore impossible on this basis alone, independent of the figure-by-figure check above. Since figures (1) and (3) were each directly confirmed achievable from the net's face layout, the box's valid views are exactly figures (1) and (3).

Only figures (1) and (3) can be formed by folding sheet (X); figures (2) and (4) require face arrangements that are geometrically impossible for this net. Hence the correct choice is '1 and 3 only'.