Rule Detection - Intermediate-Advanced Level: rule identification INTERMEDIATE-ADVANCED

PATTERN ANALYSIS:
Step 1: Track the position of the dot in each figure
- Figure 1: Position at (30, 30)
- Figure 2: Position at (50, 50)
- Figure 3: Position at (70, 70)
- Figure 4: Position at (50, 90)

Step 2: Analyze movement vectors
- Fig 1→2: Δx = 20, Δy = 20
- Fig 2→3: Δx = 20, Δy = 20
- Fig 3→4: Δx = -20, Δy = 20

Step 3: Detect movement pattern
The dot follows a diagonal movement with boundary reflection

RULE HYPOTHESIS:
Systematic positional movement following diagonal movement with boundary reflection

VERIFICATION:
All movements conform to the identified pattern ✓

APPLICATION:
Next position: (30, 70)
Following the established diagonal movement with boundary reflection

ADVANCED TECHNIQUES:
- Plot positions on coordinate system
- Check for symmetry and periodicity
- Analyze velocity and acceleration vectors
- Consider boundary conditions

COMMON MISTAKES TO AVOID:
- Assuming simple linear movement
- Not considering boundary reflections
- Missing rotational or circular patterns
- Ignoring spatial constraints

SET THEORY PATTERN ANALYSIS:

Step 1: Identify the sets
- Set A (Figure 1): Elements at specific positions
- Set B (Figure 2): Elements at specific positions

Step 2: Identify overlapping elements
- Compare positions in both sets
- Element at (50, 60) appears in BOTH sets

Step 3: Recognize the operation
- Operation symbol: −
- Operation type: Difference (A − B)

RULE HYPOTHESIS:
The rule is a Difference (A − B) operation

SET OPERATION DEFINITION:
Difference (A − B) contains elements in A that are not in B

APPLICATION:
Set A has elements: {(40, 60), (50, 60)}
Set B has elements: {(50, 60), (70, 60)}

Difference (A − B) result:
- Common elements: {(50, 60)}
- A-only elements: {(40, 60)}
- B-only elements: {(70, 60)}

Result depends on operation:
- Union: All unique = {(40, 60), (50, 60), (70, 60)}
- Intersection: Common only = {(50, 60)}
- Difference: A-only = {(40, 60)}

SET THEORY PRINCIPLES:
- Union: A ∪ B = {x | x ∈ A OR x ∈ B}
- Intersection: A ∩ B = {x | x ∈ A AND x ∈ B}
- Difference: A − B = {x | x ∈ A AND x ∉ B}

COMMON MISTAKES TO AVOID:
- Confusing union with intersection
- Forgetting to remove duplicates in union
- Wrong order in difference operation (A−B ≠ B−A)
- Miscounting common elements

PATTERN ANALYSIS:
Step 1: Identify each shape by counting sides
- Figure 1: 4 sides (quadrilateral)
- Figure 2: 3 sides (triangle)
- Figure 3: 2 sides (2-gon)
- Figure 4: 1 sides (1-gon)

Step 2: Analyze side count progression
- Fig 2 - Fig 1 = -1 sides
- Fig 3 - Fig 2 = -1 sides
- Fig 4 - Fig 3 = -1 sides

RULE HYPOTHESIS:
Each polygon loses one side in sequential transformation

VERIFICATION:
Consistent transformation: -1 side per step ✓

APPLICATION:
Figure 5: 1 - 1 = 0 sides
Shape: 0-gon

GEOMETRIC PRINCIPLES:
- Regular polygons maintain equal side lengths
- Interior angle changes with side count
- Formula: Interior angle = (n-2) × 180° / n

COMMON MISTAKES TO AVOID:
- Miscounting sides in complex polygons
- Confusing vertices with sides
- Not recognizing regular vs irregular polygons
- Missing the consistent arithmetic progression

PATTERN ANALYSIS:
Step 1: Examine the fill/shading in each figure
Step 2: Look for systematic changes in shading

RULE HYPOTHESIS:
The shading pattern alternates between filled and unfilled

VERIFICATION:
Check pattern consistency across all four figures ✓

APPLICATION:
Based on the identified rule, the next figure should continue the pattern

SHADING ANALYSIS TECHNIQUES:
- Check for binary patterns (filled/unfilled)
- Look for cyclic color patterns
- Count filled vs unfilled elements
- Observe progressive filling patterns
- Check for symmetry in shading

COMMON MISTAKES TO AVOID:
- Missing subtle shading differences
- Not recognizing cyclic patterns
- Assuming random shading changes
- Overlooking progressive patterns

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 80 units
- Figure 2: radius = 40 units
- Figure 3: radius = 20 units
- Figure 4: radius = 10 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 40 ÷ 80 = 0×
- Fig 2 → 3: 20 ÷ 40 = 0×
- Fig 3 → 4: 10 ÷ 20 = 0×

RULE HYPOTHESIS:
The circle radius halves each time (geometric progression)

VERIFICATION:
All consecutive ratios are consistent: 0× ✓

APPLICATION:
Figure 5 radius = 10 × 0 = 5 units

SCALING PATTERN TYPES:
- Geometric progression: constant r = 0

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 10 units
- Figure 2: radius = 20 units
- Figure 3: radius = 40 units
- Figure 4: radius = 80 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 20 ÷ 10 = 2×
- Fig 2 → 3: 40 ÷ 20 = 2×
- Fig 3 → 4: 80 ÷ 40 = 2×

RULE HYPOTHESIS:
The circle radius doubles each time (geometric progression)

VERIFICATION:
All consecutive ratios are consistent: 2× ✓

APPLICATION:
Figure 5 radius = 80 × 2 = 160 units

SCALING PATTERN TYPES:
- Geometric progression: constant r = 2

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 80 units
- Figure 2: radius = 40 units
- Figure 3: radius = 20 units
- Figure 4: radius = 10 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 40 ÷ 80 = 0×
- Fig 2 → 3: 20 ÷ 40 = 0×
- Fig 3 → 4: 10 ÷ 20 = 0×

RULE HYPOTHESIS:
The circle radius halves each time (geometric progression)

VERIFICATION:
All consecutive ratios are consistent: 0× ✓

APPLICATION:
Figure 5 radius = 10 × 0 = 5 units

SCALING PATTERN TYPES:
- Geometric progression: constant r = 0

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type

CONDITIONAL RULE ANALYSIS:

Step 1: Classify each figure
- Figure 1: Closed shape (circle) → has DOT inside
- Figure 2: Open shape (arc) → has LINE segment
- Figure 3: Closed shape (square) → has DOT inside
- Figure 4: Open shape (curve) → has LINE segment

Step 2: Identify the conditional pattern
Check correlation between shape type and marking:
- ALL closed shapes → contain dots ✓
- ALL open shapes → contain lines ✓

CONDITIONAL RULE HYPOTHESIS:
IF shape is CLOSED → THEN add dot inside
IF shape is OPEN → THEN add line segment

VERIFICATION:
Test hypothesis against all figures:
- Figure 1: Closed + Dot ✓
- Figure 2: Open + Line ✓
- Figure 3: Closed + Dot ✓
- Figure 4: Open + Line ✓

Rule verified across all cases ✓

APPLICATION:
Given: Next shape is a CLOSED triangle
Apply rule: IF closed → THEN add dot
Result: Triangle with dot inside

BOOLEAN LOGIC FRAMEWORK:
- Condition: IsClosed(shape)
- True branch: AddDot()
- False branch: AddLine()

CONDITIONAL RULE DETECTION STRATEGY:
1. Identify potential condition variables
2. Classify all examples by condition
3. Check for consistent outcomes per condition
4. Formulate IF-THEN rule
5. Verify rule on all examples
6. Apply to new case based on its condition

COMMON MISTAKES TO AVOID:
- Not recognizing the conditional nature
- Treating as simple alternating pattern
- Ignoring the shape property (open/closed)
- Applying wrong transformation for given condition
- Missing the IF-THEN logical structure

PATTERN ANALYSIS:
Step 1: Count elements in each figure
- Figure 1: 2 squares
- Figure 2: 3 squares
- Figure 3: 4 squares
- Figure 4: 5 squares

Step 2: Calculate differences between consecutive figures
- Fig 2 - Fig 1 = 1
- Fig 3 - Fig 2 = 1
- Fig 4 - Fig 3 = 1

RULE HYPOTHESIS:
The number of squares is increasing by 1 in each step

VERIFICATION:
All consecutive differences are consistent: +1 ✓

APPLICATION:
Figure 5 should have 5 +1 = 6 squares

COMMON MISTAKES TO AVOID:
- Miscounting elements in figures
- Not checking all consecutive differences
- Assuming non-linear progressions too early

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 35 units
- Figure 2: radius = 30 units
- Figure 3: radius = 25 units
- Figure 4: radius = 20 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 30 - 35 = -5 units
- Fig 2 → 3: 25 - 30 = -5 units
- Fig 3 → 4: 20 - 25 = -5 units

RULE HYPOTHESIS:
The circle radius decreases by 5 units (linear progression)

VERIFICATION:
All consecutive differences are consistent: -5 units ✓

APPLICATION:
Figure 5 radius = 20 + -5 = 15 units

SCALING PATTERN TYPES:
- Linear arithmetic progression: constant d = -5

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 35 units
- Figure 2: radius = 30 units
- Figure 3: radius = 25 units
- Figure 4: radius = 20 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 30 - 35 = -5 units
- Fig 2 → 3: 25 - 30 = -5 units
- Fig 3 → 4: 20 - 25 = -5 units

RULE HYPOTHESIS:
The circle radius decreases by 5 units (linear progression)

VERIFICATION:
All consecutive differences are consistent: -5 units ✓

APPLICATION:
Figure 5 radius = 20 + -5 = 15 units

SCALING PATTERN TYPES:
- Linear arithmetic progression: constant d = -5

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type

PATTERN ANALYSIS:
Step 1: Examine the fill/shading in each figure
Step 2: Look for systematic changes in shading

RULE HYPOTHESIS:
The shading pattern adds one more filled segment each time

VERIFICATION:
Check pattern consistency across all four figures ✓

APPLICATION:
Based on the identified rule, the next figure should continue the pattern

SHADING ANALYSIS TECHNIQUES:
- Check for binary patterns (filled/unfilled)
- Look for cyclic color patterns
- Count filled vs unfilled elements
- Observe progressive filling patterns
- Check for symmetry in shading

COMMON MISTAKES TO AVOID:
- Missing subtle shading differences
- Not recognizing cyclic patterns
- Assuming random shading changes
- Overlooking progressive patterns

MULTI-DIMENSIONAL PATTERN ANALYSIS:

DIMENSION 1 - Row Pattern Analysis:
Step 1: Analyze Row 1 (keeping row constant, varying column)
- Col 1: Triangle at 0°
- Col 2: Triangle at 90°
- Col 3: Triangle at 180°

Row Rule: Each column adds +90° rotation

DIMENSION 2 - Column Pattern Analysis:
Step 2: Analyze Column 1 (keeping column constant, varying row)
- Row 1: Triangle (3 sides)
- Row 2: Square (4 sides)

Column Rule: Each row adds +1 side to the polygon

RULE INTEGRATION:
For any cell[row, col]:
- Base shape determined by row (row rule)
- Rotation determined by column (column rule)

VERIFICATION:
Test on known cells:
- Cell[1,1]: Triangle + 0° ✓
- Cell[1,2]: Triangle + 90° ✓
- Cell[1,3]: Triangle + 180° ✓
- Cell[2,1]: Square + 0° ✓
- Cell[2,2]: Square + 90° ✓

APPLICATION TO MISSING CELL:
Cell[2,3] should have:
- Shape from Row 2: Square (4 sides)
- Rotation from Col 3: 180°
- Result: Square rotated 180°

MATRIX PATTERN PRINCIPLES:
- Rows often control one property
- Columns often control another property
- Cell value = f(row_property, col_property)
- Both rules apply independently

SYSTEMATIC APPROACH:
1. Identify row-wise pattern (vary column)
2. Identify column-wise pattern (vary row)
3. Verify both patterns independently
4. Combine both rules for prediction
5. Cross-verify using diagonal patterns if present

COMMON MISTAKES TO AVOID:
- Analyzing only rows or only columns
- Not recognizing independent property control
- Mixing up row and column rules
- Failing to apply both rules to prediction
- Not verifying patterns across multiple rows/columns

CONDITIONAL RULE ANALYSIS:

Step 1: Classify each figure
- Figure 1: Closed shape (circle) → has DOT inside
- Figure 2: Open shape (arc) → has LINE segment
- Figure 3: Closed shape (square) → has DOT inside
- Figure 4: Open shape (curve) → has LINE segment

Step 2: Identify the conditional pattern
Check correlation between shape type and marking:
- ALL closed shapes → contain dots ✓
- ALL open shapes → contain lines ✓

CONDITIONAL RULE HYPOTHESIS:
IF shape is CLOSED → THEN add dot inside
IF shape is OPEN → THEN add line segment

VERIFICATION:
Test hypothesis against all figures:
- Figure 1: Closed + Dot ✓
- Figure 2: Open + Line ✓
- Figure 3: Closed + Dot ✓
- Figure 4: Open + Line ✓

Rule verified across all cases ✓

APPLICATION:
Given: Next shape is a CLOSED triangle
Apply rule: IF closed → THEN add dot
Result: Triangle with dot inside

BOOLEAN LOGIC FRAMEWORK:
- Condition: IsClosed(shape)
- True branch: AddDot()
- False branch: AddLine()

CONDITIONAL RULE DETECTION STRATEGY:
1. Identify potential condition variables
2. Classify all examples by condition
3. Check for consistent outcomes per condition
4. Formulate IF-THEN rule
5. Verify rule on all examples
6. Apply to new case based on its condition

COMMON MISTAKES TO AVOID:
- Not recognizing the conditional nature
- Treating as simple alternating pattern
- Ignoring the shape property (open/closed)
- Applying wrong transformation for given condition
- Missing the IF-THEN logical structure

MULTI-DIMENSIONAL PATTERN ANALYSIS:

Transformation 1 - Rotation Analysis:
Step 1: Measure rotation angles
- Figure 1: 0°
- Figure 2: 45°
- Figure 3: 90°
- Figure 4: 135°

Rotation increment: +45° per step ✓

Transformation 2 - Size Analysis:
Step 2: Measure square dimensions
- Figure 1: 15 units
- Figure 2: 18 units
- Figure 3: 21 units
- Figure 4: 24 units

Size increment: +3 units per step ✓

COMBINED RULE HYPOTHESIS:
TWO simultaneous transformations:
1. Rotation: +45° clockwise per step
2. Scaling: +3 units per step

VERIFICATION:
Both patterns verified independently ✓
Check for correlation: None (independent transformations) ✓

APPLICATION:
Figure 5 predictions:
- Rotation: 135° + 45° = 180°
- Size: 24 + 3 = 27 units

ADVANCED MULTI-RULE DETECTION:
- Decompose complex transformations
- Analyze each dimension independently
- Verify pattern consistency for each rule
- Check for rule interactions or dependencies
- Combine predictions from all rules

COMMON MISTAKES TO AVOID:
- Focusing on only one transformation
- Missing the scaling while tracking rotation
- Not verifying both patterns independently
- Assuming transformations must be related
- Incomplete pattern analysis

PATTERN ANALYSIS:
Step 1: Count elements in each figure
- Figure 1: 6 circles
- Figure 2: 5 circles
- Figure 3: 4 circles
- Figure 4: 3 circles

Step 2: Calculate differences between consecutive figures
- Fig 2 - Fig 1 = -1
- Fig 3 - Fig 2 = -1
- Fig 4 - Fig 3 = -1

RULE HYPOTHESIS:
The number of circles is decreasing by 1 in each step

VERIFICATION:
All consecutive differences are consistent: -1 ✓

APPLICATION:
Figure 5 should have 3 -1 = 2 circles

COMMON MISTAKES TO AVOID:
- Miscounting elements in figures
- Not checking all consecutive differences
- Assuming non-linear progressions too early

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 15 units
- Figure 2: radius = 20 units
- Figure 3: radius = 25 units
- Figure 4: radius = 30 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 20 - 15 = 5 units
- Fig 2 → 3: 25 - 20 = 5 units
- Fig 3 → 4: 30 - 25 = 5 units

RULE HYPOTHESIS:
The circle radius increases by 5 units (linear progression)

VERIFICATION:
All consecutive differences are consistent: 5 units ✓

APPLICATION:
Figure 5 radius = 30 + 5 = 35 units

SCALING PATTERN TYPES:
- Linear arithmetic progression: constant d = 5

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type

PATTERN ANALYSIS:
Step 1: Examine the fill/shading in each figure
Step 2: Look for systematic changes in shading

RULE HYPOTHESIS:
The shading pattern alternates between filled and unfilled

VERIFICATION:
Check pattern consistency across all four figures ✓

APPLICATION:
Based on the identified rule, the next figure should continue the pattern

SHADING ANALYSIS TECHNIQUES:
- Check for binary patterns (filled/unfilled)
- Look for cyclic color patterns
- Count filled vs unfilled elements
- Observe progressive filling patterns
- Check for symmetry in shading

COMMON MISTAKES TO AVOID:
- Missing subtle shading differences
- Not recognizing cyclic patterns
- Assuming random shading changes
- Overlooking progressive patterns

SET THEORY PATTERN ANALYSIS:

Step 1: Identify the sets
- Set A (Figure 1): Elements at specific positions
- Set B (Figure 2): Elements at specific positions

Step 2: Identify overlapping elements
- Compare positions in both sets
- Element at (50, 60) appears in BOTH sets

Step 3: Recognize the operation
- Operation symbol: −
- Operation type: Difference (A − B)

RULE HYPOTHESIS:
The rule is a Difference (A − B) operation

SET OPERATION DEFINITION:
Difference (A − B) contains elements in A that are not in B

APPLICATION:
Set A has elements: {(40, 60), (50, 60)}
Set B has elements: {(50, 60), (70, 60)}

Difference (A − B) result:
- Common elements: {(50, 60)}
- A-only elements: {(40, 60)}
- B-only elements: {(70, 60)}

Result depends on operation:
- Union: All unique = {(40, 60), (50, 60), (70, 60)}
- Intersection: Common only = {(50, 60)}
- Difference: A-only = {(40, 60)}

SET THEORY PRINCIPLES:
- Union: A ∪ B = {x | x ∈ A OR x ∈ B}
- Intersection: A ∩ B = {x | x ∈ A AND x ∈ B}
- Difference: A − B = {x | x ∈ A AND x ∉ B}

COMMON MISTAKES TO AVOID:
- Confusing union with intersection
- Forgetting to remove duplicates in union
- Wrong order in difference operation (A−B ≠ B−A)
- Miscounting common elements

PATTERN ANALYSIS:
Step 1: Measure the radius of circles in each figure
- Figure 1: radius = 35 units
- Figure 2: radius = 30 units
- Figure 3: radius = 25 units
- Figure 4: radius = 20 units

Step 2: Calculate size changes between consecutive figures
- Fig 1 → 2: 30 - 35 = -5 units
- Fig 2 → 3: 25 - 30 = -5 units
- Fig 3 → 4: 20 - 25 = -5 units

RULE HYPOTHESIS:
The circle radius decreases by 5 units (linear progression)

VERIFICATION:
All consecutive differences are consistent: -5 units ✓

APPLICATION:
Figure 5 radius = 20 + -5 = 15 units

SCALING PATTERN TYPES:
- Linear arithmetic progression: constant d = -5

COMMON MISTAKES TO AVOID:
- Confusing diameter with radius
- Assuming linear when pattern is geometric (or vice versa)
- Miscounting the number of steps
- Not checking both differences AND ratios to identify pattern type