When the following figure is folded to form a cube, how many dots lie opposite…

Concept: When a net of six squares is folded into a cube, two faces become opposite only if they never come into contact along the fold. A quick way to find opposite pairs: in any straight run of four consecutive squares in the net, the 1st and 3rd squares fold to become opposite faces, and the 2nd and 4th squares fold to become opposite faces (each two-square gap in the row wraps exactly halfway round the cube). Faces are opposite to exactly one other face, so once two pairs are fixed, the two faces left over must be the third opposite pair.

1. The four faces bearing 1, 2, 4 and 6 dots form one straight vertical run in the net, in that top-to-bottom order.

2. Applying the alternating rule to this run of four: the face with 1 dot (1st in the run) is opposite the face with 4 dots (3rd in the run).

3. Likewise, the face with 2 dots (2nd in the run) is opposite the face with 6 dots (4th in the run).

4. That leaves only the face with 3 dots (attached beside the 2-dot face) and the face with 5 dots (attached beside the 6-dot face); since a cube has exactly three opposite pairs and the other two pairs are already fixed, these two must be opposite each other.

Cross-check: This can be confirmed by physically folding the net: hold the face with 2 dots flat as the base. The face with 1 dot folds up to become one wall, and the face with 4 dots folds up on the opposite side of the base, confirming it lands directly across from the 1-dot wall. Continuing the same strip, the face with 6 dots folds twice to close up as the top, landing directly above the 2-dot base — confirming 2 and 6 are opposite. The face with 3 dots, attached to the base's remaining free edge, becomes a third wall; the face with 5 dots, attached past the top face, wraps around to land directly across from the 3-dot wall — confirming that the face bearing three dots lies opposite the face bearing five dots.

Hence, when the figure is folded to form a cube, the face opposite the one bearing five dots carries three dots.