Choose the box that is similar to the box formed by folding the given sheet of…

CONCEPT: When a flat net is folded into a cube, two faces attached to each other along a shared edge become ADJACENT faces of the cube, never opposite faces. Two golden sub-rules solve cube-net questions without physically folding: (1) In a straight run of four squares (a row or a column), the 1st and 3rd squares always end up opposite each other, and the 2nd and 4th squares always end up opposite each other, because the four squares wrap around into a closed loop of side faces. (2) A flap folded out perpendicular to one end square becomes adjacent to that square; two such perpendicular flaps coming off different ends of the same run form the third pair of opposite faces. Consequently, any valid 3-face view of a folded cube must show exactly ONE face from each of the three opposite-face pairs -- never two faces from the same pair, since two opposite faces of a cube can never be seen together in a single view.

APPLICATION: Label the four squares running down the column of sheet (X), top to bottom, as A (blank), B (blank), C (the circle face) and D (blank); the flap attached to A is one half-shaded face and the flap attached to D is the other half-shaded face.

1. By the column rule, the 1st and 3rd squares of the run are opposite, so A is opposite C.

2. By the same rule, the 2nd and 4th squares of the run are opposite, so B is opposite D.

3. The two half-shaded flaps come off opposite ends of the same column, so they form the third pair: the two half-shaded faces are opposite each other.

4. The three opposite pairs are therefore {A, C}, {B, D}, and {the two half-shaded faces}.

5. Cube (1): top blank, one side shaded, one side blank -- exactly one face from each pair. Valid.

6. Cube (2): top shows the circle face (C), but both visible side faces are blank -- this forces the two visible sides to be B and D together, yet B and D are themselves an opposite pair and can never both be visible at once. Invalid.

7. Cube (3): top blank, one side shaded, one side blank -- again exactly one face from each pair. Valid.

8. Cube (4): both visible side faces are half-shaded -- the two half-shaded faces (an opposite pair) are shown together, which is impossible. Invalid.

CROSS-CHECK: Every valid folding must use exactly three faces, one per opposite pair. Cubes (1) and (3) are the only two candidates that do -- they differ only in which of the two remaining blank faces and which of the two half-shaded faces they show, consistent with two genuinely different valid orientations of the same net. Cubes (2) and (4) each place a same-pair face twice, so neither can be formed from sheet (X).

Hence only the boxes in figures (1) and (3) can be formed by folding sheet (X).