Cube & Dice Reasoning: Tricks & Examples

Cube and Dice look visual and scary, but a few formulas and rules handle almost every question. PAINTED CUBE (cut into n×n×n smaller cubes) • Corner cubes (3 faces painted): always 8. • Edge cubes (2 faces painted): 12 × (n − 2). • Face-centre cubes (1 face painted): 6 × (n − 2)². • Inner cubes (0 faces painted): (n − 2)³. Example for n = 3: corners 8, edges 12, one-face 6, none 1 (total 27). DICE • Opposite faces never touch. On a standard dice, opposite faces add up to 7 (1-6, 2-5, 3-4). • Two dice positions: if two views share a common face, the faces that change tell you which are adjacent (so the remaining one is opposite). SOLVED EXAMPLE A cube painted on all sides is cut into 64 small cubes (n = 4). • 2 faces painted (edges) = 12 × (4−2) = 24. • 0 faces painted (inner) = (4−2)³ = 8. EXAM ANGLE: For painted cubes, just plug n into the four formulas. For dice, find the opposite face using the "sum 7" or the common-face method. Practise cube & dice sets on Govt Exam Center.