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 80 cm is painted yellow on all the faces and then cut into smaller cubes of sides 8 cm each. Find the number of smaller cube having all the three faces painted.
A cube of side 49 cm is painted purple on all the faces and then cut into smaller cubes of sides 7 cm each. Find the number of smaller cubes having only one face printed.
A cube of side 18 cm is painted yellow on all the faces and then cut into smaller cubes of sides 3 cm each. Find the number of smaller cubes that have only two faces painted.
A cube of side 12 cm is painted green on all the faces and then cut into smaller cubes of sides 2 cm each. Find the number of smaller cubes that have only one face painted.
Four friends J, Q, B and Z rolled the dice in alphabetical order.
The scores were:
1st round: 3, 2, 5, 2
2nd round: 1, 5, 1, 4
3rd round: 4, 2, 1, 1
If each point on the dice would get 10 points, who won the maximum points after 3 rounds?
