Dice Puzzle Strategy
To find the number opposite a specific face on a die, we analyze the relationships between the faces shown in different positions.
Analyzing Dice Positions
We are given three different views of the same die:
- Position 1: Faces showing 6, 1, 4
- Position 2: Faces showing 2, 3, 1
- Position 3: Faces showing 6, 4, 5
The goal is to find the number on the face opposite the face showing '4'.
Finding Opposite Faces Using Common Elements
Compare Position 1 (6, 1, 4) and Position 2 (2, 3, 1):
- The common face between these two positions is '1'.
- When one face is common, we can find opposite faces by considering the faces adjacent to the common face in a consistent rotational order (e.g., clockwise).
- In Position 1, starting from '1' and moving clockwise (relative to the die's orientation), the adjacent faces shown are 4 and 6. Let's represent the sequence as 1 -> 4 -> 6.
- In Position 2, starting from the common face '1' and moving clockwise, the adjacent faces shown are 3 and 2. Let's represent the sequence as 1 -> 3 -> 2.
- By aligning the common face '1' and comparing the subsequent faces in order, we can determine opposite pairs. The face '4' (from Position 1's sequence) is opposite the face '3' (from Position 2's sequence). Similarly, '6' is opposite '2'.
Confirming Opposite Pairs
We can verify this using another pair of positions. Compare Position 1 (6, 1, 4) and Position 3 (6, 4, 5):
- The common faces between Position 1 and Position 3 are '6' and '4'.
- A rule for dice puzzles states: If two faces are common in two different positions of the same die, then the remaining faces must be opposite to each other.
- In Position 1, the face remaining after excluding '6' and '4' is '1'.
- In Position 3, the face remaining after excluding '6' and '4' is '5'.
- Therefore, '1' is opposite '5'. This confirms our understanding of the die's configuration.
The derived pairs of opposite faces are (4, 3), (6, 2), and (1, 5).
Identifying the Number Opposite '4'
From the analysis using the common face '1' between Position 1 and Position 2, we concluded that '4' is opposite '3'.
The number on the face opposite the face showing '4' is 3.
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?
Three different positions of the same dice are shown. Select the number that will be on the face opposite to the face having the number '3'?
Select the dice that can be formed by folding the given sheet along the lines.
