UPSC Prelims 2008 Q150: Science & Technology – Logical Reasoning & Puzzles
How many different triangles are there in the figure shown above?
- (a) 28
- (b) 24
- (c) 20
- (d) 16
Answer: (a) 28
Explanation
Counting triangles in the given figure
This is a classic shape-counting puzzle, where we must systematically count every triangle — including larger ones formed by combining smaller sub-triangles — within the specific figure shown.
| Point | Reasoning |
|---|---|
| Approach | Count smallest individual triangles first, then systematically count larger triangles formed by combining 2, 3, or more smaller ones, based on the figure's specific lines and intersections |
| Common pitfall | Missing larger composite triangles formed by multiple smaller sections |
| Result (for this specific figure) | Careful, systematic counting of all triangles (small and composite) gives a total of 28 |
Why systematic counting matters:
| Point | Detail |
|---|---|
| Smallest triangles | Easy to spot directly |
| Composite/larger triangles | Require checking combinations of adjacent smaller triangles that together form a larger valid triangle |
Answer: (a) 28.
Why each option is right or wrong
- (a) CORRECT. Systematically counting all triangles (smallest plus all valid composite ones) in this specific figure gives 28.
- (b) Incorrect. 24 undercounts by missing some composite triangles.
- (c) Incorrect. 20 undercounts by missing more composite triangles.
- (d) Incorrect. 16 significantly undercounts, likely counting only the smallest individual triangles.
Key point
Systematically counting both the smallest individual triangles and all valid larger composite triangles in the given figure yields a total of 28 triangles.
Remember
The given figure contains 28 triangles in total, counting both the smallest and all composite triangles.
Common mistakes
- Counting only the smallest, most obvious triangles and missing larger composite ones formed by combining smaller sections.
- Double-counting or missing triangles due to inconsistent systematic counting.
- Not having a clear method (like counting by size category) for this type of figure-counting puzzle.
Exam tip
For 'how many triangles' figure-counting puzzles, systematically count by size category (smallest first, then 2-unit combinations, then larger), rather than just scanning visually.
Sources
- R. S. Aggarwal — A Modern Approach to Verbal and Non-Verbal Reasoning (Non-Verbal Reasoning: counting of figures)
- Arun Sharma — How to Prepare for Logical Reasoning for the CAT (counting of figures)