Rules (questions 61–75)
- PAT: 90 questions, 60 minutes, 6 sections of 15 in a fixed order; about 40 seconds per question or 10 minutes per section on an even pace. No penalty for guessing.
- A figure of identical cubes cemented together is painted on every exposed side EXCEPT the bottom it rests on.
- Assume hidden cubes only where they are needed to support visible cubes.
- Several questions share one figure: 'How many cubes have exactly N sides painted?' Answers run from 1 to 5 cubes.
Method: one pass, one tally
- Sketch the footprint as a grid and write each column's height (count hidden supporting cubes).
- Make a tally with rows 0–5 painted sides.
- Go column by column, top cube to bottom cube, using a fixed face order (top, front, back, left, right).
- For each cube: painted sides = (1 if nothing sits on it) + number of its 4 side faces not touching another cube. The bottom face never counts.
- Check: the tallies must add to the total number of cubes (sum of column heights). Then answer every question for that figure from the same tally.
Patterns worth knowing
| Figure | Painted-side counts |
|---|---|
| Single cube on the floor | 1 cube with 5 |
| Two-cube stack | Top 5, bottom 4 |
| Row of 3 on the floor | Ends 4, middle 3 |
| Solid 2 × 2 × 2 block | 4 top cubes with 3, 4 bottom cubes with 2 |
| Solid 3 × 3 × 3 block (27 cubes) | 4 with 3, 12 with 2, 9 with 1, 2 with 0 |
| Staircase in a row: heights 3, 2, 1 | One with 5, two with 4, two with 3, one with 2 (6 cubes) |
- 5 painted sides only on a cube that sits higher than all its neighbors (or stands alone).
- A middle cube in a tall stack has its top and bottom covered: at most 4 painted sides.
- Cubes touching only along an edge or a corner do not cover each other's faces.
⚠️ Common traps
- Counting the bottom face as painted.
- Forgetting hidden supporting cubes (often the 0, 1 or 2 painted-side cubes).
- Answering 'two or more' when the question asks for exactly two.
- Recounting the figure for each question instead of reusing the tally, which invites drift.
Mini-quiz
- What is the maximum number of painted sides any cube can have?
- Three cubes stand in a row on the floor with nothing on top. How many painted sides does the middle cube have?
- A column of three cubes stands alone. How many painted sides does the middle cube have?
- A solid 3 × 3 × 3 block rests on the floor. How many cubes have exactly 2 painted sides?
- A row of three columns has heights 3, 2 and 1. How many cubes have exactly 4 painted sides?
Answer key: 1 C · 2 C · 3 C · 4 C · 5 D