Dice Opposite Faces: Understanding Adjacency
This problem involves understanding the spatial relationships between the faces of a standard six-sided die. We are given specific pairs of opposite faces and need to determine which faces are adjacent to a particular face.
Analyzing the Given Dice Information
We are told:
- Face 1 is opposite to Face 4.
- Face 2 is opposite to Face 5.
A standard die has six faces numbered from 1 to 6. Since we know the pairs (1, 4) and (2, 5) are opposite, we can deduce the remaining opposite pair.
- The faces are {1, 2, 3, 4, 5, 6}.
- Known opposite pairs: (1, 4), (2, 5).
- The remaining faces are 3 and 6. Therefore, Face 3 must be opposite to Face 6.
So, the complete set of opposite faces based on the problem statement is:
- 1 is opposite 4
- 2 is opposite 5
- 3 is opposite 6
Determining Adjacent Faces
On a standard die, any given face is adjacent to all faces except the one directly opposite it.
Let's find the faces adjacent to the numbers mentioned in the options:
- Adjacent to 1: All faces except 4 are adjacent. These are {2, 3, 5, 6}.
- Adjacent to 2: All faces except 5 are adjacent. These are {1, 3, 4, 6}.
- Adjacent to 3: All faces except 6 are adjacent. These are {1, 2, 4, 5}.
- Adjacent to 4: All faces except 1 are adjacent. These are {2, 3, 5, 6}.
- Adjacent to 5: All faces except 2 are adjacent. These are {1, 3, 4, 6}.
- Adjacent to 6: All faces except 3 are adjacent. These are {1, 2, 4, 5}.
Evaluating the Options
Now, let's check each option against our derived adjacent faces:
-
3 is adjacent to 4 and 6
- Is 3 adjacent to 4? Yes (Faces adjacent to 3 are {1, 2, 4, 5}).
- Is 3 adjacent to 6? No (Face 3 is opposite to Face 6).
Therefore, this statement is incorrect.
-
3 is adjacent to 5 and 6
- Is 3 adjacent to 5? Yes (Faces adjacent to 3 are {1, 2, 4, 5}).
- Is 3 adjacent to 6? No (Face 3 is opposite to Face 6).
Therefore, this statement is incorrect.
-
2 is adjacent to 3 and 6
- Is 2 adjacent to 3? Yes (Faces adjacent to 2 are {1, 3, 4, 6}).
- Is 2 adjacent to 6? Yes (Faces adjacent to 2 are {1, 3, 4, 6}).
Therefore, this statement is correct.
-
1 is adjacent to 4 and 6
- Is 1 adjacent to 4? No (Face 1 is opposite to Face 4).
- Is 1 adjacent to 6? Yes (Faces adjacent to 1 are {2, 3, 5, 6}).
Therefore, this statement is incorrect.
Conclusion
Based on the analysis, the statement that correctly describes the adjacency of faces on the die, given the provided opposite pairs, is that 2 is adjacent to 3 and 6.
Important Questions from Cube and Dice
A cube of side 4 cm is painted green on all its faces and then divided into smaller cubes of side 2 cm each. How many small cubes have been obtained?
Two adjacent faces of a solid cube are painted with black colour. The faces opposite to the black faces is painted with blue colour while the remaining faces are painted with yellow colour. After painting, this cube has been divided into 125 equal cubes. How many cubes have no faces painted?
A cube, whose two adjacent faces are coloured, is cut into 64 identical small cubes. How many of these small cubes are not coloured at all?
Two different positions of the same dice are shown, the six faces of which are marked with the letters A to F. Select the letter that will be on the face opposite to the one having A.
A Cube is coloured blue on all the faces. It is then cut into 64 smaller cubes. These cubes are now separated. The number of smaller cubes with no surface coloured is:
