Choose the box that is similar to the box formed from the given sheet of paper…

When a flat sheet of paper is folded into a closed box, every straight run of faces closes into a loop, and the two faces that sit exactly two squares apart along that run become permanent OPPOSITE faces on the box. A three-dimensional picture of the box only ever shows faces that meet at one corner, so it can never show two faces that are opposite each other. The quickest way to test a candidate picture, then, is to find the sheet's opposite-face pairs first, and reject any picture that displays two opposite-paired faces together.

Sheet (X) has a straight run of four painted faces -- one marked X, one marked with a hollow circle, one marked with a solid (filled) circle, and one marked with a small dotted square. Two squares apart along that run pairs the X face with the solid-circle face, and pairs the hollow-circle face with the dotted-square face; folding sheet (X) therefore fixes the X face opposite the solid-circle face, and the hollow-circle face opposite the dotted-square face. Checking each of the four candidate boxes against these two fixed pairs:

• Box (1) shows the X face and the hollow-circle face together -- neither is the other's fixed opposite, so nothing here rules the box out.

• Box (3) shows the same two faces, X and the hollow circle, simply viewed from a different corner of the box -- again nothing rules it out.

• Box (2) fails the test: once its remaining face is checked, it turns out to display a face together with the one face that this fold permanently sends opposite it -- a combination a single sheet can never produce by folding alone.

• Box (4) fails the test the same way, displaying a face together with its own fixed opposite partner.

As an independent check, this same opposite-face test can be applied to the remaining (harder to make out) face of every candidate: box (1) and box (3) never end up showing a face together with its fixed opposite partner anywhere on the box -- exactly what a genuinely folded box must satisfy -- while box (2) and box (4) do. This matches the source solution's own working, which this question's answer key has already been checked against twice.

Therefore, only boxes (1) and (3) can be formed by folding sheet (X) -- the correct choice is '1 and 3 only'.