SATHEE: Cube Dice Rules - Quick Revision

Cube Dice Rules - Quick Revision

Opposite faces never appear together in any dice position.
Sum of opposite faces in a standard dice = 7 (1-6, 2-5, 3-4).
Adjacent faces share an edge; if two faces are opposite, they can’t be adjacent.
In an open dice (net), opposite faces are never in the same row/column (gap rule).
Clockwise/Counter-clockwise rule: Rotate dice mentally to track face movements.
Common edge = common number: If two positions share an edge, the number on that edge is fixed.
Eliminate options where opposite faces appear together in single-dice questions.
In two-dice comparison, mark common faces first to deduce opposite pairs.
If a number is on top, its opposite is on the bottom (hidden face).
In cube painting, (n-2)³ gives inner cubes without paint (for n×n×n cube).
Cubes with 3 painted faces = 8 corners; 2-painted faces = 12(n-2) edges; 1-painted face = 6(n-2)² centers.
Dice with non-standard numbers: Check opposite sums ≠7 first.
In unfolded dice, cross-check T-junctions for correct adjacency.
Always draw a mini-map of adjacent/opposite faces for complex dice.
Practice 3D visualization daily—use pen as axis to rotate mentally.

Formula/Rule	Application
Opposite Sum = 7	Standard dice: 1↔6, 2↔5, 3↔4.
Inner Cubes (unpainted) = (n-2)³	For n×n×n painted cube.
3-face painted = 8	Corner cubes always.
2-face painted = 12(n-2)	Edge cubes excluding corners.
1-face painted = 6(n-2)²	Face-center cubes.
0-face painted = (n-2)³	Deep inner cubes.
Gap Rule in Net	Opposite faces have 1 face gap in cross-shaped net.
Clockwise Rotation	Track face movement: right rotation → top face moves right.
Common Face Rule	If two dice show same adjacent pair, third face is opposite.
Elimination Rule	Remove options violating opposite/adjacent rules.

“7-UP”: Opposite faces sum to 7—think of the soft drink.
“C-C-C”: Corners = 8, Count by (n-2) for edges/centers.
“T-Junction”: In nets, T-shape junction confirms adjacency, not opposition.
“Right-Hand Rule”: Stick right thumb on top face; fingers show rotation direction.
“GAP = OPP”: In cross-net, 1-gap faces are opposite.

Mistake	Correct Approach
Assuming all dice follow sum=7	Check for non-standard dice first.
Ignoring hidden face	Remember bottom face is opposite to top.
Miscounting painted cubes	Apply (n-2) formulas systematically.
Confusing adjacent vs opposite	Use net diagrams to verify.
Overlooking rotation direction	Always track clockwise/counter-clockwise changes.

1. Which number is opposite 4 in a standard dice?

> **Answer: 3** (Sum 7 rule)

2. How many cubes have exactly 2 faces painted in a 4×4×4 cube?

> **Answer: 24** (12(n-2) = 12×2)

3. In the given net, which face is opposite to face 2?

> **Answer: 5** (Gap rule in cross-net)

4. If top=1, front=3, right=5, what’s on the bottom?

> **Answer: 6** (Opposite of 1)

5. How many inner cubes are unpainted in a 5×5×5 cube?

> **Answer: 27** ((5-2)³ = 27)