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Analytical Reasoning 3% exam weight

Cube and Dice Problems

Part of the HAT-UG (HEC Aptitude Test - Undergraduate) study roadmap. Analytical Reasoning topic ar-15 of Analytical Reasoning.

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Cube and Dice Problems

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A standard die is a six-faced cube whose opposite faces always sit in fixed pairs: (1, 6), (2, 5), (3, 4). Every opposite pair sums to 7, and this identity never changes under rotation. The task in HAT-UG questions is to read a 2-D picture of two or three visible faces and decide which face is opposite (or which face appears on a specific side after a rotation).

RuleStatement
Opposite pair ruleFaces 1–6, 2–5, 3–4 are opposite; sum = 7
Two-position ruleIf two faces appear together in two different positions, they are opposite
Adjacency ruleFaces that share an edge in any view are adjacent, never opposite
  • Track rotation by imagining a clock face fixed on the top of the cube.
  • Eliminate choices that list two adjacent faces as opposite — they are traps.
  • The die has 6 faces, 12 edges, and 8 vertices; each vertex touches 3 adjacent faces.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Standard Die Geometry

A conventional die obeys a strict numbering convention so that the answer to any opposite-face question is mechanical. The pairs (1, 6), (2, 5), (3, 4) are invariant under any rotation of the cube. Because rotation is a rigid motion, the relative position of every face stays the same — only the observer’s viewpoint changes. Any HAT-UG item that prints numbered faces on three visible sides is therefore solvable by inspection: pick the two numbers that are not adjacent in the picture and check whether they are an opposite pair.

The Two-Position Adjacency Rule

When a question shows two different positions of the same die and asks for the opposite face, apply this rule: if a pair of faces appears together in both positions, they must be opposite, because no two adjacent faces can stay together in two independent rotations of the same cube. If the pair only appears together once, treat them as adjacent and resolve the rest of the cube by elimination.

Clockwise and Anticlockwise Rotation

Rotation problems give a starting view and a final view and ask whether the die is turned clockwise or anticlockwise around a fixed axis (usually the vertical axis through the top face). Compute the direction as seen from the top — a movement that matches the rotation of clock hands is clockwise.

  • Pick the top face as the axis of rotation; it does not move.
  • Trace the rotation of the front face to the right side.
  • If the front face moves to the right, the die turned clockwise; to the left means anticlockwise.
Face shown in view AFace shown in view BConclusion
1 and 3 together1 and 3 together1 and 3 are opposite
2 and 4 together2 and 5 together2 is adjacent to 4, opposite to 5
4, 5, 6 together4, 5, 6 togetherImpossible — three faces of one cube cannot repeat exactly

Stacking and Counting Cubes

For an n × n × n arrangement of identical cubes, the total number of cubes is n³. The number of cubes visible on the surface is 6n² − 12n + 8, since the inner cubes are hidden. Count layer by layer: the base layer is always full, and each upper layer sits directly on the cubes below it, so hidden cubes = (n − 2) × (n − 2) × n for the middle slabs.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Examiner Traps

The most common HAT-UG trap prints two adjacent faces in a single view and offers the answer “they are opposite.” Reject it: the two-position rule requires two independent positions, not a single picture. Another trap reverses the direction of rotation when the cube is viewed from the bottom rather than the top. Always fix the observer above the top face before deciding clockwise or anticlockwise.

Shading and Face-Colour Problems

When a cube is painted on three faces that meet at one vertex, the three opposite faces remain unpainted. This is the standard “three-face shaded” rule and appears verbatim in HAT-UG. A cube has 8 vertices, and each vertex is the meeting point of exactly 3 mutually adjacent faces — so the rule pivots on identifying the painted vertex.

Worked Example

A die shows 2 on top, 3 on the front, 5 on the right. Which face is opposite to 3?

  • 3 is adjacent to 2 (top) and 5 (right) in this view.
  • The opposite pair for 3 is always 4.
  • Verify: 4 must be on the bottom, because bottom is opposite to top, and 4 is also adjacent to 2 and 5 in the view. Both conditions hold, so 4 is opposite to 3.

A second die shows 1–2 together in position P and 1–2 together in position Q. The two-position rule forces 1 and 2 to be opposite, which contradicts the standard pair (1, 6). Resolve by re-reading the question: the die is non-standard, so the only useful fact is that 1 and 2 are opposite, and the remaining pairs (4, 5) and (3, 6) must be inferred from any third face shown.

  1. A die shows faces 4, 5, 6 with 4 on top. After a 90° clockwise rotation around the vertical axis, which face sits on the front? (Answer: 5.)
  2. A cube is painted on three faces meeting at a vertex, then cut into 27 smaller cubes. How many small cubes have exactly two painted faces? (Answer: 12 — the edge cubes between painted and unpainted faces.)

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