Cubes, Dice and Painted Cubes — The Four Counting Formulas

Cubes and dice look like a drawing question, so candidates try to picture the cube in their head, lose the far side, and guess. It is not a drawing question. It is four formulas and two rules, and a student who knows them answers in fifteen seconds without imagining anything.

The setup never changes. A big cube of side n is painted on all six faces, then sliced into n³ unit cubes. Every unit cube now sits in exactly one of four places: a corner, an edge, a face, or buried inside. Where it sits decides how many painted sides it kept — 3, 2, 1 or 0. That is the whole question.

Dice questions are a separate two-rule system built on one fact: on a standard dice, opposite faces add to 7. SSC, RRB, CTET and BPSC TRE-4 all set two to four marks here every cycle, and the marks are close to free — provided you stop visualising and start counting.

1. A cube of side 4 cm is painted on all its faces and then cut into 1 cm cubes. How many of the small cubes have exactly two faces painted?

2. In the same 4 cm painted cube cut into 1 cm cubes, how many small cubes have no face painted at all?

3. A painted cube is cut into 125 identical smaller cubes. How many of them have exactly one face painted?

4. What is the minimum number of straight cuts required to divide a cube into 64 identical smaller cubes?

5. Two positions of the same dice show the faces 1, 2, 3 and 1, 4, 5. Which number lies opposite 1?

6. On a standard dice the faces 1, 2 and 3 meet at one corner. What is the sum of the three faces meeting at the diagonally opposite corner?

7. A solid block measuring 5 × 4 × 3 unit cubes is painted on all its outer surfaces and then separated into unit cubes. How many unit cubes have no painted face?

8. Two positions of a dice show top 2, front 3, right 1 and top 2, front 3, right 5. Which pair of numbers lies on opposite faces?