Cubes and dice

In short

Cubes and dice questions test three-dimensional visualisation on paper: deciding which faces of a die lie opposite one another from two or more flat views, folding a net into a cube, and counting how many small cubes carry paint after a big painted cube is sliced. The whole topic rests on one geometric fact — a cube has three pairs of opposite faces, and no two faces of a pair can ever be seen together. It is standard in reasoning papers of SSC CGL/CHSL, RRB NTPC, IBPS and SBI PO/Clerk, CDS, NDA, AFCAT and state PSC exams, and in campus aptitude rounds (TCS NQT, Infosys, Wipro, Capgemini, Accenture, AMCAT, CoCubes, eLitmus), where two or three such questions are near-certain. The same spatial skill underpins engineering graphics, 3D CAD and carton/pallet stacking, so it is also a quick screen for visualisation ability in design and operations interviews.

Explained

The cube and the die: facts everything is built on

• A cube has 6 faces, 12 edges, 8 vertices; the 6 faces form 3 pairs of opposite (parallel) faces; every face is adjacent (perpendicular) to exactly 4 faces and opposite to exactly 1.
• Any single view of a cube (drawn in isometric style) shows at most 3 faces, and those three meet at one vertex — so they are mutually adjacent and come one from each opposite pair. This single sentence solves most dice questions.
• A cube has 24 rotations, so numbering its faces 1-6 gives 6!/24 = 30 distinguishable dice.
• Ordinary (common) die: opposite faces sum to 7 — pairs (1,6), (2,5), (3,4); total pips = 1+2+3+4+5+6 = 21.
• Standard die: an ordinary die whose faces 1, 2, 3 run anticlockwise about their common corner (with 1 on top and 2 towards you, 3 is on the right). The mirror image, with 1-2-3 clockwise, is also sold; most Indian aptitude books assume the anticlockwise convention, so a well-set question gives enough views and you should not import the chirality rule unless the paper says "standard dice".
• Only two dice satisfy the 7-sum rule — the two mirror forms. The faces may carry dots, numbers, letters, colours or symbols; the logic is identical.

Dice: getting the opposite pairs out of the views

• Rule 1 (adjacency): numbers seen together in one view are adjacent, never opposite. On an ordinary die, two numbers summing to 7 can never appear in the same view.
• Rule 2 (two common numbers): if two views share two numbers, the two odd numbers out are opposite. Reason: each view takes one face from each pair, so the shared two use up two pairs and the remaining two must both belong to the third pair. Positions do not matter.
• Rule 3 (exhaustion): if, across the views, a number X is seen together with four different numbers, the fifth remaining number is opposite X.
• Rule 4 (one common number, four distinct others): views {X, a, b} and {X, c, d} with a, b, c, d distinct give the same conclusion as Rule 3 — the number never shown is opposite X.
• Rule 5 (rotation): moving top → front → right → top (a cyclic shift of the three visible numbers) is a genuine 120° rotation about the body diagonal, so such views describe the same die. Swapping only two of the three visible numbers is a reflection and is impossible for one die — that is how "which view cannot belong to the same die?" is answered.
• Fix the pairs first, then answer whatever is asked (sum, product, "which face lies below", "which cannot be adjacent").
• Hidden-pip questions: for a vertical tower of k ordinary dice, top + bottom of each die = 7; hidden pips = 21k − visible pips.

Nets: folding and unfolding a cube

• A net is the cube opened flat; a cube has exactly 11 distinct nets. Faces sharing an edge in the net are adjacent on the cube.
• Skip-one rule: in a straight row or column, two faces with exactly one face between them are opposite. A straight strip of four wraps round the cube as a band — faces 1 and 3, and 2 and 4, are opposite.
• Corner rule: two squares touching only at a corner (diagonally placed) are adjacent, not opposite — a frequent trap.
• Staircase (zigzag) rule: in a six-square zigzag strip, faces three steps apart along the strip are opposite (verified by rolling: steps E, S, E bring you to the opposite face).
• Each step along a net equals one 90° roll of the cube; two rolls in the same direction give 180° (opposite face), two rolls in perpendicular directions give an adjacent face.
• Folding questions ("which cube can be made from this net?") are answered by testing the options against the opposite pairs: any option showing two opposite faces together is out; survivors are then checked for the sense of rotation of the symbols.
• Unfolding questions run the other way: given a cube with symbols, mark the pairs, then reject nets that place a pair adjacent.

Cut and painted cubes: the counting formulas

• A cube of side n (in units of the small cube) sliced completely gives n³ unit cubes and needs (n − 1) cuts in each of three directions, i.e. 3(n − 1) cuts.
• With all six faces painted and n ≥ 2: corners carry 3 painted faces (8 cubes), edge-interiors carry 2 (12(n − 2)), face-interiors carry 1 (6(n − 2)²) and the core carries none ((n − 2)³).
• Check your answer two ways: the four counts add to n³, and the painted unit faces 8×3 + 12(n−2)×2 + 6(n−2)² add to 6n², the surface area in unit squares.
• "At least one face painted" = n³ − (n − 2)³ = 6n² − 12n + 8; "at least two" = 8 + 12(n − 2) = 12n − 16.
• Different colours: if each pair of opposite faces gets its own colour, corner cubes show 3 colours, edge cubes 2, face cubes 1. If two adjacent faces share a colour, the edge between them produces cubes with two painted faces of the same colour — count them separately.
• Partial painting (some faces left unpainted, or a face painted in two colours) has no formula worth memorising. Use the coordinate method: index each small cube (i, j, k) with 1 ≤ i, j, k ≤ n; it is painted on a given side exactly when its coordinate is at that extreme and that side is painted. Count by cases or by inclusion-exclusion.
• Unequal blocks (cuboid l × b × h): 3-face = 8, 2-face = 4[(l−2) + (b−2) + (h−2)], 1-face = 2[(l−2)(b−2) + (b−2)(h−2) + (h−2)(l−2)], 0-face = (l−2)(b−2)(h−2). Any bracket that turns negative is treated as 0 and the figure is counted directly.

Cuts, stacks and visible-cube counting

• Maximum pieces from c cuts: split c as a + b + d (planes in three perpendicular directions) and multiply (a+1)(b+1)(d+1) — largest when the three are as equal as possible. Pieces then need not be equal in size; "identical cubes" forces 3(n − 1) cuts.
• Minimum cuts for n³ identical cubes cannot be reduced by stacking slices unless the question allows re-arranging between cuts (some papers do; read the wording).
• Visible/invisible cubes in an n-cube: visible from outside = n³ − (n − 2)³; completely hidden = (n − 2)³ (for 3×3×3: 26 visible, 1 hidden).
• Stacked-figure counting (an irregular pile drawn in isometric view): count layer by layer from the bottom, and remember that cubes hidden behind the drawing must be inferred from the fact that a cube cannot float — supports must exist beneath it.
• Volume and surface relations are occasionally asked: cutting raises total surface area without changing volume; a cube of side n cut into unit cubes has total surface 6n³ against the original 6n², so the newly exposed (unpainted) area is 6n³ − 6n².

Non-standard variants that appear in papers

• Dice with letters/colours: identical logic; answers ask for "the colour opposite red".
• Two dice shown touching: the touching faces are hidden and are not opposite each other — they belong to different dice; use the 7-sum rule on each die separately.
• Dice-as-data-sufficiency: "Is 4 opposite 3?" with two statements; decide whether the views pin down the third pair — they do when two views share exactly two numbers, when a number is seen with four others, or when the pairs already fixed leave only one choice.
• Cube rolled along a path (direction-sense flavour): track the bottom face through each roll; four rolls in the same straight direction restore the original orientation.
• Painted cube then re-cut unevenly, or outer layer removed: the remaining solid is a cube of side (n − 2) with no paint — a one-line answer once seen.

Exam approach

1. Read whether the die is "ordinary/standard" — if yes, write the pairs 1-6, 2-5, 3-4 immediately and check the views for consistency.
2. Otherwise list the views as sets, hunt for two shared numbers (Rule 2) or a number seen with four others (Rule 3), and write the three pairs before reading the options.
3. For painted cubes, write n, then the four formulas; verify the total is n³.
4. For nets, mark the three pairs on the net using the skip-one rule, then eliminate options. Budget 30-45 seconds per question; a question that needs more is better left for the review pass.

Key points

• Three pairs - a cube has exactly 3 pairs of opposite faces; one view shows one face from each pair.
• Never together - faces of the same pair can never be seen in one view; on an ordinary die, 1&6, 2&5, 3&4 never co-appear.
• 7-sum rule - on an ordinary die the face opposite x is 7 − x; total pips 21.
• Standard die - ordinary die with 1, 2, 3 anticlockwise round their shared corner; the mirror form also exists.
• Two-common rule - if two views share two numbers, the remaining two numbers are opposite, whatever their positions.
• Exhaustion rule - a number seen with four different numbers is opposite the fifth, unseen number.
• Cyclic shift - top→front→right→top is a legal rotation; swapping just two visible numbers is impossible for one die.
• Skip-one - in a net, faces with one face between them in a row or column are opposite.
• Corner-touching - diagonally placed net squares are adjacent, not opposite.
• Zigzag three-step - in a six-square staircase net, faces three steps apart are opposite.
• 11 nets - a cube unfolds in exactly 11 distinct ways.
• Slicing - n³ unit cubes need 3(n − 1) cuts; c face-parallel cuts give at most (a+1)(b+1)(d+1) pieces with a + b + d = c.
• Painted counts - 8 three-face, 12(n − 2) two-face, 6(n − 2)² one-face, (n − 2)³ unpainted.
• Two checks - counts sum to n³ and painted unit faces sum to 6n².
• Unpainted core - removing the painted skin leaves a cube of side n − 2.
• Cuboid - 3-face count stays 8; the others use (l−2), (b−2), (h−2) separately.
• Partial painting - no formula; use coordinates (i, j, k) and count cases.
• Tower of dice - hidden pips = 21k − visible pips, since top + bottom = 7 for each die.

Formulas and examples

Example 1 - painted cube, side 5. A 15 cm cube is painted on all faces and cut into 3 cm cubes.

• n = 15/3 = 5, so 5³ = 125 small cubes, needing 3(5 − 1) = 12 cuts.
• Three faces painted = 8; two faces = 12 × 3 = 36; one face = 6 × 3² = 54; none = 3³ = 27.
• Check: 8 + 36 + 54 + 27 = 125 ✓. Painted unit faces: 24 + 72 + 54 = 150 = 6 × 5² ✓.
• At least one face painted = 125 − 27 = 98.
• If instead only 7 cuts (all parallel to the faces) were allowed, maximum pieces = 3 + 2 + 2 → 4 × 3 × 3 = 36 (against 4 × 4 × 2 = 32 or 6 × 2 × 2 = 24).

Example 2 - opposite faces from views, and a dice tower. (a) Three views of a die show: I → 1 (top), 2 (front), 3 (right); II → 1 (top), 4 (front), 5 (right); III → 2 and 4 together.

• From I and II, 1 is adjacent to 2, 3, 4, 5 → 6 is opposite 1.
• From III, 2 is adjacent to 4; 2 is already adjacent to 1 and 3, so 2 can only pair with 5, leaving 3 opposite 4.
• Pairs: (1,6), (2,5), (3,4) — each sums to 7, so this is an ordinary die. The face opposite 5 is 2.

(b) Three such dice are stacked vertically and the topmost face shows 2. Hidden faces are the two touching faces at each of the two junctions and the tower's base — five faces. Hidden pips = (7 − 2) + 7 + 7 = 19. Equivalently, total 63 minus the visible 44.

Example 3 - net to cube. A net has a row of four squares A, B, C, D (left to right), with E above B and F below B.

• Skip-one in the row: A opposite C, B opposite D; in the column E-B-F: E opposite F.
• So A is adjacent to B, D, E, F — everything except C.
• If the numbers were A = 1, B = 2, C = 3, D = 4, E = 5, F = 6, the pairs are (1,3), (2,4), (5,6) and the sums are 4, 6, 11 — not an ordinary die, so the 7-sum rule must not be used on it.
• A folded option showing 1, 3 and 5 together is impossible (1 and 3 are opposite); an option showing 1, 2, 5 is geometrically possible and must then be checked for the sense of rotation.

Example 4 - cuboid. A 6 cm × 4 cm × 3 cm block is painted all over and cut into 1 cm cubes (72 pieces).

• 3 faces = 8; 2 faces = 4[(6−2) + (4−2) + (3−2)] = 4 × 7 = 28; 1 face = 2[(4)(2) + (2)(1) + (1)(4)] = 2 × 14 = 28; 0 faces = 4 × 2 × 1 = 8.
• Check: 8 + 28 + 28 + 8 = 72 ✓. Painted unit faces: 24 + 56 + 28 = 108 = 2(6×4 + 4×3 + 3×6) ✓.

Example 5 - partial painting. A cube of side 4 is painted only on its four vertical (side) faces; top and bottom are left bare. It is cut into 64 unit cubes.

• Three painted faces: 0 — a corner cube touches two side faces plus a bare face.
• Two painted faces: the four vertical edges, 4 cubes each → 16.
• One painted face: each vertical face has 16 cubes, minus the 8 lying on the vertical edges → 8 per face × 4 = 32.
• No paint: the inner 2 × 2 column through all 4 layers → 16. Check: 0 + 16 + 32 + 16 = 64 ✓.

Where it is used

• Campus aptitude rounds: TCS NQT, Infosys, Wipro Elite, Capgemini, Accenture, Cognizant and the AMCAT/CoCubes/eLitmus tests all carry cube-and-dice items in the logical section, usually 1-3 questions.
• Government and defence recruitment: SSC CGL/CHSL/MTS, RRB NTPC and Group D, IBPS and SBI PO/Clerk, LIC AAO, CDS, NDA, AFCAT and state PSC prelims — non-verbal reasoning blocks where dice and folded-paper figures sit together.
• Engineering graphics and CAD: the same visualisation is tested in first-year engineering drawing (orthographic and isometric views, development of surfaces) and used daily in SolidWorks/CATIA/AutoCAD modelling, sheet-metal unfolding and FEM meshing.
• Logistics and packaging: carton and pallet stacking, container cube utilisation and warehouse slotting at e-commerce and 3PL firms are cube-counting problems with rupee consequences per trip.
• Manufacturing and QC roles: reading a component from 2D views, jig and fixture orientation, and specifying which faces are machined or coated — the industrial version of "which faces are painted".
• Product and game design: dice and board-game components (Ludo, snakes and ladders), Rubik's-cube style puzzles, and the handedness convention that makes a mirrored die a different product.

Interview questions

• Q: Why can two faces seen in the same view never be opposite? A: The three faces in a view meet at one vertex, so they are mutually perpendicular; opposite faces are parallel and cannot share a vertex.
• Q: On an ordinary die, what lies opposite 4, and how do you know? A: 3, because opposite faces of an ordinary die sum to 7 — the pairs are (1,6), (2,5) and (3,4).
• Q: Two views of a die show 2, 3, 5 and 2, 3, 6. Which faces are opposite? A: 5 and 6 are opposite, since both views share 2 and 3, so the odd faces out must come from the third pair.
• Q: A number is seen alongside four different numbers in several views. What follows? A: It is adjacent to those four, so the only remaining number is opposite it.
• Q: Why are 1-2-3 anticlockwise and 1-2-3 clockwise dice different objects? A: They are mirror images; no rotation converts one into the other, so a question showing both orders for the same die is inconsistent.
• Q: How many small cubes have exactly two faces painted when a painted cube of side n is cut into n³ cubes? A: 12(n − 2) — the cubes along the 12 edges, excluding the 8 corners.
• Q: How many unit cubes stay completely unpainted, and what shape do they form? A: (n − 2)³, forming a cube of side n − 2 at the core.
• Q: What is the maximum number of pieces from 6 straight cuts of a cube, all parallel to its faces? A: 27 — split as 2 + 2 + 2 giving 3 × 3 × 3; pieces need not be equal in size.
• Q: A cube is painted red on two opposite faces and green on the rest, then cut into 27. How many have both colours? A: The red faces contribute 9 cubes each; of those, the 8 non-central cubes on each face also touch a green face, so 16 carry both.
• Q: In a net, two squares touch only at a corner. Are those faces adjacent or opposite? A: Adjacent — only squares with exactly one square between them in a straight line are opposite.
• Q: Three ordinary dice are stacked and the top face shows 3. What is the sum of the hidden pips? A: 4 + 7 + 7 = 18, since each die's top and bottom sum to 7 and the top die's bottom is 7 − 3.
• Q: How would you handle a painted-cube question where only three faces meeting at a corner are painted? A: Drop the formulas and use coordinates: a unit cube's painted-face count equals how many of its three coordinates sit at a painted extreme; count the cases.

Common mistakes

• Applying the 7-sum rule to a die whose given faces contradict it, or to letter/colour dice — check consistency before assuming "ordinary".
• Treating two numbers seen together as opposite, or concluding "opposite" from a single view, which never determines any pair.
• Insisting that the two shared numbers in Rule 2 occupy the same positions; the deduction works for the view sets, positions irrelevant.
• Reading corner-touching squares in a net as opposite faces, and forgetting that a straight strip of four wraps round the cube.
• Using 12(n − 2) or 6(n − 2)² when some faces are unpainted, when adjacent faces share a colour, or when the solid is a cuboid.
• Forgetting the cross-check — the four painted counts must total n³ and the painted unit faces must total 6n²; most arithmetic slips surface here.
• Confusing "minimum cuts for identical cubes" (3(n − 1)) with "maximum pieces from c cuts", and assuming cut pieces must be equal.

Related topics

Non-verbal reasoning: figures and patterns · Direction sense · Seating arrangement and puzzles · Floor, box and scheduling puzzles · Analogies and classification · Data sufficiency