Cubes and Dice is one of the most predictable scoring areas in AFCAT Reasoning and Military Aptitude. Once you memorise a handful of cube-cutting formulas and the two golden rules of dice, these questions become near-instant marks. This Cavalier guide turns a tricky-looking spatial topic into pure plug-and-play arithmetic.
Why Cubes and Dice Matter in AFCAT
The AFCAT Reasoning and Military Aptitude section blends verbal and non-verbal reasoning, and spatial-visualisation questions like Cubes and Dice appear almost every cycle. They typically come as a quick set of 2–4 questions, and they reward candidates who have a system over those who try to visualise everything from scratch.
The good news: unlike many reasoning topics, this one is formula-driven. There is very little ambiguity. If you know the rule, you get the mark in under 30 seconds — a huge advantage in a time-pressured paper. Compare that with a tricky seating-arrangement or syllogism question that can swallow two or three minutes and still leave you unsure of the answer.
Because AFCAT has a fixed time budget and negative marking, the smart play is to bank these easy spatial marks early and save your thinking time for tougher series and arrangement questions. A confident candidate clears a cube-dice set without scratch-work; that confidence comes only from drilling the formulas until they are automatic.
There is a deeper reason the Air Force tests spatial ability: officers constantly read three-dimensional information from two-dimensional maps, displays and instruments. So while revising this topic you are also building exactly the visualisation skill the service values — treat it as both an exam chapter and a slice of officer-like aptitude training.
Treat Cubes and Dice as “free marks.” Lock in the formulas now, and never spend more than a minute on any single question. If a dice picture looks unusually complex, mark it for review and return after securing the easy ones.
Cube Basics You Must Know
A cube has 6 faces, 12 edges and 8 corners (vertices). Every face is a square, and opposite faces are parallel.
In AFCAT, the most common cube question is the painted-and-cut cube. A large cube is painted on all outer faces, then sliced into smaller identical cubes. You are asked how many small cubes have a certain number of painted faces.
The key variable is n — the number of cuts per edge, i.e. how many small cubes fit along one edge of the big cube. If a cube is cut into 64 small cubes, then n3 = 64, so n = 4.
Almost every cube question hinges on first finding n correctly. Because the total is always a perfect cube, you only need to recognise a small set of perfect cubes by heart: 8, 27, 64, 125, 216 and 343. Spotting which one the question uses is half the battle, and it removes any need for long division or trial multiplication during the exam.
Total small cubes = n3. So find n by taking the cube root of the total. n = 2 → 8 cubes, n = 3 → 27, n = 4 → 64, n = 5 → 125.
The Four Painted-Cube Formulas
When a fully painted cube is cut into n3 smaller cubes, the small cubes fall into exactly four categories. Memorise these — they are the heart of the topic.
- 3 painted faces (corner cubes): always 8 (the 8 corners).
- 2 painted faces (edge cubes): 12 × (n − 2).
- 1 painted face (centre-of-face cubes): 6 × (n − 2)2.
- 0 painted faces (inner cubes): (n − 2)3.
A quick sanity check: add all four results and you should get exactly n3. If they don’t add up, you’ve slipped somewhere.
The number of corner (3-face) cubes is always 8, no matter how big the cube is — because a cube always has 8 corners.
Why the Formulas Work (Quick Logic)
You don’t have to take the formulas on faith. The logic is simple and helps you recall them under pressure.
Corners get 3 colours
Each corner cube sits where three painted faces meet, so it shows paint on 3 sides. A cube has 8 corners → 8 cubes.
Edges get 2 colours
Along each of the 12 edges (excluding the two corner cubes at its ends) the small cubes touch two painted faces. That leaves (n − 2) cubes per edge → 12(n − 2).
Face-centres get 1 colour
On each of the 6 faces, ignore the outer ring; the inner (n − 2) × (n − 2) block touches only that one painted face → 6(n − 2)2.
Inner cubes get no paint
The hidden core, an (n − 2) cube on every side, never touched paint → (n − 2)3.
For n = 2 (cut into 8), every small cube is a corner cube: all 8 have 3 painted faces and the other categories are 0.
Worked Example: Painted Cube
A cube painted red on all faces is cut into 125 equal small cubes. How many small cubes have (a) exactly 2 red faces and (b) no red face?
So 36 cubes have exactly 2 red faces and 27 have no painted face. Check: 8 + 36 + 6(9) + 27 = 8 + 36 + 54 + 27 = 125. ✓
Two-Colour and Partial-Paint Variations
Examiners sometimes paint different faces with different colours, or leave one pair of opposite faces unpainted. The base formulas still apply, but you must read carefully.
One pair of faces unpainted
If, say, the top and bottom are not painted, then any small cube touching only those faces gets fewer painted faces than the standard formula predicts. Re-count corners and edges that border the unpainted faces.
Different colours on adjacent faces
Counting cubes with “exactly red and green” means counting edge cubes lying on the shared edge of those two coloured faces only.
Don’t blindly apply 12(n − 2) when faces are partly unpainted or multicoloured. The formula assumes all six faces painted in one colour. Adjust by re-counting the affected edges and corners.
Dice Fundamentals
A standard dice is a cube with faces numbered (or marked) 1 to 6. On an ordinary/standard dice, the numbers on opposite faces sum to 7: 1–6, 2–5, 3–4.
Dice questions usually show the same dice in two or more positions and ask which number lies opposite a given number, or which number is on top/bottom. In AFCAT the faces may carry dots, letters, shapes or colours instead of plain digits, so train yourself to apply the same logic to any markings, not just numbers.
- Common-face rule: If two views share the same (common) face in the same position, the other faces in those views are not opposite to it — the remaining two unseen faces are opposite each other.
- Standard-dice rule: When told it is a standard dice, opposite faces always add to 7.
Finding Opposite Faces: The Common-Face Method
This is the workhorse method for AFCAT dice questions where the dice is not stated to be standard. It works purely from the two or more pictures the question gives you, so you never have to assume the 1–6, 2–5, 3–4 pairing.
If one face is common in both views
Suppose the face showing X appears in the same position in two views. Then X is fixed. The two faces that change position between the views are not opposite to X — so the two faces you can’t see must be opposite to those visible ones.
If two faces are common
When two faces stay the same across both views, the third visible faces in each view are opposite to each other.
List the three faces visible in each view. Any number that appears adjacent to a target number in any view can never be opposite it. Eliminate adjacents, and the survivor is the opposite face.
Worked Example: Dice
Two positions of the same dice are shown. View 1 shows faces 1, 2, 3 (with 2 on top). View 2 shows faces 4, 2, 3 (again 2 on top). Which number is opposite 1?
Therefore the number opposite 1 is 4. (The remaining pair 2 and ... and 3 and 6 / 5 fill the rest depending on the full net.)
A face seen adjacent to the target in any single view is automatically ruled out as its opposite.
Open Dice and Nets
Some questions give an open dice (net) — the flattened cross/T shape — and ask which faces become opposite when folded, or which of four given nets can fold into a shown dice.
The trick is not to mentally fold the paper. Instead, classify each pair of faces using a simple positional rule. This converts a confusing 3-D visualisation task into a flat, 2-D inspection you can do at a glance.
- Faces that are directly next to each other in the net are adjacent (never opposite).
- Faces separated by exactly one face in a straight line become opposite when folded.
- An L-shaped pair (turning a corner) becomes adjacent, not opposite.
So in a straight strip of four faces, the 1st and 3rd are opposite, and the 2nd and 4th are opposite.
Students wrongly assume touching faces in a net are opposite. Touching = adjacent. Only the “one-gap-in-a-line” faces are opposite.
Common Traps and How to Beat Them
AFCAT setters reuse the same handful of traps. Know them and you’ll never lose a mark to carelessness.
- Standard vs ordinary dice: Only use the “sum to 7” rule when the question says standard. Otherwise use the common-face method.
- Mis-reading n: n is cubes per edge, not total cubes. Cube root first.
- Forgetting the check-sum: Painted-cube answers must add up to n3.
- Adjacent ≠ opposite: In nets and views, touching faces are adjacent.
- Partial painting: Re-count when faces are unpainted or multicoloured.
- Rushing the picture: Read whether the question asks for “at least one” painted face versus “exactly one.” “At least one” means total cubes minus the inner unpainted cubes.
Most errors here are reading errors, not concept errors. Slow down for one second on the wording, then apply the rule mechanically. The candidates who lose marks are almost always the ones who solved the right formula for the wrong question.
If a dice question gives three or more views, focus on the view pair sharing the most common faces — it cracks the puzzle fastest.
Previous-Year Style Practice
Q. A cube is painted blue on all faces and then cut into 64 identical small cubes. How many of these small cubes have exactly one face painted blue?
Answer: Total = 64, so n³ = 64 → n = 4. One-face cubes = 6 × (n − 2)2 = 6 × (4 − 2)2 = 6 × 4 = 24 cubes.
- Total small cubes = n3; find n by cube root.
- 3 faces = 8; 2 faces = 12(n−2); 1 face = 6(n−2)2; 0 faces = (n−2)3.
- All four must sum to n3 — always check.
- Standard dice: opposite faces sum to 7.
- Otherwise use the common-face rule; adjacent faces are never opposite.
- In nets, faces one gap apart in a line are opposite.
Frequently asked questions
How many small cubes have no painted face when a cube is cut into n cubes per edge?
The unpainted inner cubes number (n − 2)3. For example, a cube cut into 64 pieces has n = 4, giving (4 − 2)3 = 8 unpainted cubes.
How do I know if a dice is standard?
The question must state it is a standard or ordinary dice. Only then do opposite faces sum to 7 (1-6, 2-5, 3-4). If it is not stated, use the common-face method from the given views.
What is the difference between adjacent and opposite faces on a dice?
Adjacent faces share an edge and can be seen together in one view; opposite faces are parallel and can never appear in the same view. Any face seen next to your target is automatically not its opposite.
How many cubes always have three painted faces?
Exactly 8, regardless of cube size, because these are the corner cubes and every cube has 8 corners.
How should I read an open dice (net) question?
Faces touching in the net are adjacent. Faces separated by exactly one face along a straight line become opposite when the net is folded; faces turning an L-corner stay adjacent.
How much time should Cubes and Dice take in AFCAT?
With the formulas memorised, each question should take 20 to 40 seconds. They are among the fastest scoring items in the Reasoning and Military Aptitude section.
Related AFCAT Reasoning and Military Aptitude topics
Want a teacher to walk you through AFCAT Reasoning and Military Aptitude?
Cavalier's AFCAT batches break every topic into classroom sessions with daily practice, tests and doubt-clearing.