Karnataka Police SI – Reasoning Ability
Figure Series (Pattern Recognition)
Goal – Identify the rule that governs a sequence of geometric figures and predict the next figure or the missing one.
1. What is a Figure Series?
A figure series is a set of two or more geometric shapes arranged in a linear order (horizontal or vertical).
Each figure is derived from the previous one by applying one or more systematic transformations (size, orientation, colour, number of elements, symmetry, etc.).
Exam tip – Treat it as a visual puzzle where the “hidden variable” can be:
- Number of sides / vertices
- Shape type (circle, square, triangle, etc.)
- Size / scale
- Colour or shading
- Number of repetitions of a sub‑figure
- Orientation (rotation/reflection)
- Symmetry axis
- Position of components (top, bottom, left, right)
2. Common Types of Figure Series
| # | Pattern Type | Typical Features | Example |
|---|---|---|---|
| 1 | Size / Scale | Each figure grows or shrinks uniformly. | A circle → an enlarged circle → a further enlarged circle |
| 2 | Rotation | Same shape rotated by a constant angle each step. | Triangle → 90° rotation → 180° rotation … |
| 3 | Reflection | Mirror image about a vertical/horizontal axis. | Square → reflected → reflected again |
| 4 | Addition / Subtraction of Elements | A new shape is added or removed at each step. | Square → square + small circle → square + two circles |
| 5 | Colour Change | Colour of a particular part changes systematically. | Red circle → blue circle → green circle |
| 6 | Symmetry Change | Number of symmetry axes changes. | 4‑symmetric figure → 2‑symmetric figure |
| 7 | Composite Transformation | Combination of two or more of the above (e.g., enlarge + rotate). | Square → rotate 45° and enlarge |
3. Systematic Approach to Solve
| Step | Action | Why It Helps |
|---|---|---|
| 1. Identify the base figure | Look at the first figure; note shape, size, colour. | Establishes the starting point. |
| 2. Count discrete elements | Vertices, sides, circles, squares, etc. | Helps spot arithmetic or geometric progression. |
| 3. Observe orientation changes | Note angles of rotation; direction (clockwise/counter‑clockwise). | Detects rotational patterns. |
| 4. Look for symmetry | Count axes of symmetry or symmetry type (horizontal, vertical, rotational). | Reveals symmetry progression. |
| 5. Check colour or shading | Note any systematic change. | Colour often changes in a cycle. |
| 6. Notice addition/subtraction | Count how many elements appear/disappear. | Detects additive patterns. |
| 7. Compile all observations | Write a short hypothesis (e.g., “size increases by 20 % and rotates 90° each step”). | Combines the clues into a rule. |
| 8. Verify with all figures | Apply the rule to each figure; if all fit, it’s likely correct. | Validates the hypothesis. |
| 9. Predict the missing figure | Use the rule to generate the next figure. | Final answer. |
4. Common Pitfalls & Tricks
| Pitfall | How to Avoid |
|---|---|
| Misidentifying the “base” | Always start with the first figure, not the one that looks simplest. |
| Assuming linear scale change | Check if the scale factor is constant or varies (e.g., 1.5×, then 2×). |
| Ignoring colour cycles | Colour may repeat every 3 or 4 figures; check for cycles. |
| Missing composite transformations | A figure may both enlarge and rotate; test both possibilities. |
| Over‑interpreting noise | Random-looking details are usually irrelevant; focus on consistent changes. |
| Failing to test all figures | A rule that fits only 2 or 3 figures is suspect; test all. |
5. Key “Formulas” & Relationships
| Concept | Formula / Relation | Use |
|---|---|---|
| Area of regular polygon | (A = \frac{1}{4} n s^2 \cot\frac{\pi}{n}) (n sides, side length s) | When size change is by scaling, area changes by the square of the scale factor. |
| Scale factor | (k = \frac{\text{new side}}{\text{old side}}) | Detects uniform scaling. |
| Rotation angle | ( \theta_{n} = \theta_{1} + (n-1)\Delta\theta ) | Linear rotation sequence. |
| **Symmetry |