Cubes and dice questions test spatial reasoning: visualizing how faces meet, identifying opposite faces, and tracking a solid after it is rotated. The drawing is only a clue; use the face relationships it proves.

1. Basic cube facts

• A cube has 6 faces, 12 edges, and 8 vertices.
• Each face has four adjacent faces and exactly one opposite face.
• Three faces meet at every vertex; opposite faces can never meet.
• When a cube is rotated, face positions change but adjacency and opposite pairs do not.

2. Reading dice positions

Two dice views that share a common face can reveal which faces are adjacent. Keep the common face fixed mentally and compare the faces that appear around it. Faces that occur together in any view are adjacent; a pair never shown together may be opposite once the other four adjacencies are known.

• Standard die rule: opposite faces add to 7: 1-6, 2-5, and 3-4.
• Common-face rule: if the same face appears in two views, the remaining visible faces are arranged around it.
• Rotation rule: never assume left and right stay fixed after the die is turned.

3. Open cubes and nets

An open cube is a net: a two-dimensional arrangement of six squares that folds into a cube. Imagine lifting one square at a time through a right angle. A face cannot become adjacent to a square that lies directly across the folded cube.

1. Choose a central square as the base face.
2. Fold the squares directly attached to it upward to form side faces.
3. Identify the remaining square as the face opposite the base.
4. Check a proposed pair: if the faces would touch along an edge, they are adjacent.

4. Painted cube counting

When a large cube is painted on all outer faces and cut into $n x n x n$ smaller cubes, classify each small cube by its position. This is quicker and more reliable than trying to count individual cubes from a diagram.

• Three faces painted: 8 corner cubes.
• Two faces painted: $12(n - 2)$ edge cubes, excluding corners.
• One face painted: $6(n - 2)^2$ face-center cubes.
• No face painted: $(n - 2)^3$ interior cubes.
• Total: the four categories add up to $n^3$.

A reliable solving method

1. Mark every pair of faces that is visibly adjacent.
2. Use the fact that each face has only one opposite face to eliminate options.
3. For dice, focus on the common face rather than comparing screen positions.
4. For nets, fold from a chosen base face and test only the pair asked about.
5. For painted cubes, decide whether the small cube is a corner, edge, face-center, or interior cube.

Common mistakes

• Treating two faces as opposite simply because they look far apart in a flat net.
• Comparing the left side of two dice drawings without accounting for rotation.
• Including corner cubes in the count of cubes with exactly two painted faces.
• Forgetting that the formulas for painted cubes require n to be at least 2.

Quick revision summary

• Adjacent faces meet; opposite faces never meet.
• On a standard die, opposite numbers sum to 7.
• Use a common face to compare two dice positions.
• For painted cubes, classify by position before applying a formula.