A dice is numbered from 1 to 6 in different ways. If 1 is adjacent to 2, 4 and…

Concept: On a standard six-sided die, every face has exactly one face opposite to it and four faces adjacent to it. Any two faces on the die are always either that single opposite pair or an adjacent pair -- there is no third possibility.

Applying it here: Face 1 is already adjacent to 2, 4 and 6, which fills three of its four adjacent slots. Only the numbers 3 and 5 remain unplaced relative to 1.

1. Face 1 needs exactly one more adjacent face (its fourth) and exactly one opposite face, and only 3 and 5 are left to fill those two roles.

2. So one of {3, 5} must be the single face opposite 1, and the other must be its fourth adjacent face.

3. Since only one face can be opposite 1, and that role is already taken by either 3 or 5, the two of them -- 3 and 5 -- cannot be opposite each other.

4. By the concept above, two faces that are not opposite must be adjacent -- so 3 and 5 are necessarily adjacent to each other, regardless of which one ends up opposite 1.

Cross-check: Try both cases directly -- if 3 sits opposite 1, then 5 is adjacent to 1 and to 3; if 5 sits opposite 1, then 3 is adjacent to 1 and to 5. In either case 3 and 5 end up adjacent, confirming the result holds independent of the exact layout.

Result: 3 is adjacent to 5.