Opposite-face rules for standard and non-standard dice, painted-cube cutting problems, and systematic counting of triangles/lines in a figure.
📖 ~14 min read🧩 SSC Reasoning
Introduction
This final chapter covers three related visual-counting topics: determining opposite faces
of a dice from given views, calculating how many smaller cubes get painted when a larger painted cube is
cut, and systematically counting triangles/lines/squares in a given figure.
Standard Dice — Opposite Face Rule
Diagram — Dice Net Showing Opposite Face Pairs
Unfolded dice net: when folded into a cube, "2" (top row, when folded) sits opposite "4" in a standard row; "5" and "6" (top/bottom flaps) become opposite faces. In an unfolded net, faces separated by exactly one face in a straight line become opposite once folded.
Rule
In a net, two squares are opposite faces if exactly one square lies between them in a straight row/column of the unfolded net
To find the opposite of a face, use two different given views of the dice sharing a common face, and eliminate
Q. In two positions of a dice, the number 1 is adjacent to 2, 3, and 4 in one view, and adjacent to 2, 4, and 5 in another. What is opposite to 1?
Numbers adjacent to 1 across both views: 2, 3, 4, 5 — that's 4 distinct faces adjacent to 1
A cube has 6 faces; 1 itself is one face, its 4 neighbours are four more → the remaining face, 6, is opposite to 1
Painted Cube — Cutting Formulas
Diagram — A Painted Cube Cut into 27 Smaller Cubes (3×3×3)
A 3×3×3 cube's front face shown — corner cubes get 3 painted faces, edge cubes get 2, face-centre cubes get 1, and the fully internal cube gets 0.
Position (in an n×n×n cube, all 6 outer faces painted)
Formula (count of such small cubes)
3 faces painted (corners)
Always 8 (regardless of n, for n≥2)
2 faces painted (edges, not corners)
12 × (n−2)
1 face painted (face centres)
6 × (n−2)²
0 faces painted (fully internal)
(n−2)³
Q. A cube is painted on all faces and cut into 64 smaller equal cubes. How many small cubes have exactly 2 painted faces?
💡 Systematic Method: Label every distinct intersection point in the figure, then count (1) the smallest individual shapes first, (2) shapes made by combining 2 small units, (3) shapes made by combining 3 or more units — proceeding systematically size-by-size avoids missing or double-counting any figure.
Q. A triangle is divided into 4 smaller triangles by joining the midpoints of its sides. How many total triangles are there in the figure?
4 smallest triangles (from the subdivision) + 1 large original triangle = 5 triangles
✅ Practice Focus: Dice net opposite-face rule and the "eliminate adjacent faces" method for open-view dice · Painted-cube formulas (8 corners always; 12(n−2) edges; 6(n−2)² faces; (n−2)³ internal) · Size-by-size systematic counting for triangles/lines/squares.