Four positions of a dice are shown below. What number must be at the bottom…

On a die, two faces seen together in the same sketch are ADJACENT, and can never be opposite each other. A face is confirmed OPPOSITE another only when every sketch that face appears in has been checked and all of its other adjacent faces are accounted for — leaving exactly one face that never appeared with it; merely being absent from a single sketch is not enough evidence on its own.

Applying this to the dice shown:

1. List the three visible faces in each sketch: (i) 5, 4, 6; (ii) 4, 1, 5; (iii) 3, 6, 5; (iv) 2, 1, 4.

2. Track every face that appears alongside 4: sketch (i) pairs it with 5 and 6; sketch (ii) pairs it with 1 and 5; sketch (iv) pairs it with 2 and 1.

3. Collecting these shows 4 is adjacent to 5, 6, 1, and 2 — four of the five other faces on the die.

4. The only number never seen with 4 is 3, so 3 and 4 must be the die's remaining opposite pair.

5. In sketch (iii), 3 is on top, so its opposite face, 4, occupies the bottom.

Since a die's six faces split into exactly three opposite pairs, checking the remaining faces confirms consistency: 1 never appears with 6, and 2 never appears with 5 — matching a valid pairing of 1-6 and 2-5 that leaves 3-4 as the only pair possible, exactly matching the result above, with each opposite pair summing to 7 as on a standard die.