Two related question types share this chapter: finding which face lies opposite another on a die shown from different angles, and counting how many small cubes get painted when a larger cube is cut apart. No question from this topic appeared in the papers I could extract or recover, so this chapter is built entirely to the AAI pattern.
A cube always has three pairs of opposite faces. Every question in this chapter is really about pinning down those three pairs, or about counting how many of many small cubes touch the outer surface in a particular way.
A standard die is numbered 1 to 6 and follows one universal rule: opposite faces always add up to seven. This rule applies only to a die explicitly described as standard, never to a die carrying letters, symbols, or colours.
Every pair adds to seven. This is the single fact to know for a standard die, and nothing more needs to be deduced from any picture.
The sum-to-seven rule is the most misapplied rule in this whole chapter. If the die shows letters or symbols instead of numbers 1 to 6, or if the question does not say "standard die," this rule does not apply at all, and the opposite faces must be worked out from the views given.
When a die carries letters or symbols, its opposite faces are not fixed by any universal rule. Instead, they must be worked out from however many different views the question shows.
A cube shows at most three faces at once, and those three all meet at one corner, so none of them can be opposite any other. Every pair you see together in one picture rules that pair out as a possible opposite pair.
This is a direct consequence of the rule above: since the two common faces are already known to be adjacent to everything else shown, the only faces left unexplained are the two that switched — and the only relationship two faces can have on a cube, if not adjacent, is opposite.
Across all the given views, write down every pair of faces that ever appear together. Those pairs are all adjacent, never opposite. With three views covering all six faces, elimination alone often reveals all three opposite pairs without needing the two-common-faces shortcut at all.
A die with faces A, B, C, D, E and F is shown in two positions. Position 1 shows A on top, B in front, and C on the right. Position 2 shows A on top, B in front, and D on the right. What can be deduced?
A large cube is painted on all six outer faces, then cut into many identical smaller cubes. The question asks how many of the small cubes have a given number of painted faces.
These four counts always add up to exactly n³, the total number of small cubes, which is a quick way to check your working.
Every cube, whatever its size, has exactly 8 corners. The other three formulas all depend on how much of the cube's edge length remains once the two ends touching a painted face are set aside, which is where the (n − 2) term comes from.
A cube is painted on all faces and then cut into 64 identical smaller cubes. How many of the smaller cubes have exactly two painted faces?
Practice covering the standard dice rule, opposite faces from multiple views, and painted cube cutting with worked numeric answers. Print it, solve it, keep it.
Multiple choice questions, building in difficulty. Full solutions are in the downloadable PDF above.
Live classes cover every reasoning, aptitude, English and GK topic straight from the real question papers, with the shortcut for each question type drilled until it is automatic.