Two positions of the same dice are shown below. The three visible faces in…

On any die, however its six faces happen to be numbered, the three pairs of opposite faces are fixed: turning, flipping or rolling the die never changes which value sits opposite which. Two faces that can be seen together in a single view therefore share an edge and can never be opposite each other. If one value occupies the same visible face in two different views, the die has merely been turned about the axis through that face; such a turn leaves that face in place and cycles the other four faces in a fixed cyclic order (top to front-left to bottom to rear-right and back to top). That cyclic order is an invariant of the die, and it is what lets both views be merged into one complete picture.

Applying this to the two given positions:

1. The value 3 sits on the front-right face in both position (i) and position (ii), so 3 is the pivot: the die has only been turned about the axis through 3 and the face opposite it.

2. Everything seen alongside 3 must be adjacent to 3. Position (i) shows 1 and 6 with it, and position (ii) shows 2 and 5 with it. That is four different faces adjacent to 3, so the only value left over, 4, has to be the face opposite 3.

3. The four values 1, 6, 2 and 5 therefore form the ring of faces around the pivot axis. Read that ring in the fixed spatial order top, front-left, bottom, rear-right: position (i) reads 1, 6, ?, ? and position (ii) reads 2, 5, ?, ?.

4. A turn about the pivot axis can only shift the ring cyclically, so the two readings must be cyclic shifts of one and the same 4-cycle. Position (i) can only be completed as 1, 6, 2, 5 or as 1, 6, 5, 2. In the cycle 1 to 6 to 2 to 5, the value 2 is followed by 5, exactly as position (ii) requires. In the cycle 1 to 6 to 5 to 2, the value 2 is followed by 1, which contradicts position (ii).

5. So the ring runs 1 to 6 to 2 to 5. In position (i) this puts 2 on the bottom face and 5 on the rear-right face, completing the die.

6. Opposite faces lie two places apart in the ring, which pairs the top face 1 with the bottom face 2, and the front-left face 6 with the rear-right face 5.

Cross-check the completed die against both views:

Every value from 1 to 6 is used exactly once and no view shows two opposite values together, so the arrangement is consistent. Since 1 and 2 are an opposite pair, when 2 is at the bottom the number at the top is 1.