Two positions of a dice are shown below. How many points will be on the top…

Concept: When two views of the same dice share one common face in the same position (same side, same orientation), the die has simply rotated about the axis running through that fixed face and its opposite. The remaining two visible faces (top and front) cycle around this axis; if none of the top/front values from the first view repeats among the top/front values of the second view, the rotation between the views is a half turn (180 degrees), so each face and its counterpart in the SAME slot (top-to-top, front-to-front) across the two views are opposite faces.

1. In the first position, the fixed face (3 dots) sits on the right; the top shows 2 dots and the front shows 6 dots.

2. In the second position, the same fixed face (3 dots) sits on the right, in the same orientation; the top now shows 1 dot and the front shows 5 dots.

3. Since 3 stays fixed and neither 2 nor 6 reappears among {1, 5}, the rotation between the two views is a half turn about the left-right axis, so top pairs with top and front pairs with front: 2 is opposite 1, and 6 is opposite 5.

4. The only number not yet placed is 4, so by elimination 4 must be opposite the fixed face, 3.

5. Since 2 is opposite 1, placing 2 points at the bottom brings 1 point to the top.

Cross-check: The four numbers besides the fixed face - 1, 2, 5, 6 - pair up without conflict as (2, 1) and (6, 5), leaving (3, 4) as the only remaining pair; every face has exactly one opposite and no number repeats, confirming the pairing is consistent.