Rule Detection - Beginner Level: figure rules BEGINNER

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

VERIFICATION:
All consecutive differences are consistent: +1 ✓

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

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

ABSTRACT RELATIONSHIP ANALYSIS:

Step 1: Analyze the relationship between Figure 1 and Figure 2
- Figure 1: Triangle
- Figure 2: 3 circles

Step 2: Identify properties of Figure 1
- Shape: Triangle
- Number of corners/vertices: 3
- Number of sides: 3

Step 3: Identify properties of Figure 2
- Shape: Circles
- Count: 3

RELATIONSHIP HYPOTHESIS:
The count of shapes in Figure 2 equals the number of corners in Figure 1

Step 4: Formulate the rule
Rule: "The second figure in each pair contains as many shapes as the first figure has corners"

VERIFICATION:
- Figure 1: Triangle has 3 corners
- Figure 2: Shows 3 circles ✓
- Relationship confirmed!

APPLICATION TO NEW PAIR:
- Figure 3: Square has 4 corners
- Figure 4 should: Show 4 shapes (circles)

ABSTRACT REASONING PRINCIPLES:
- Look beyond geometric transformations
- Consider numerical relationships
- Examine property mappings (corners → count)
- Test hypothesis on known pairs
- Apply verified rule to new cases

RELATIONSHIP DETECTION STRATEGIES:
1. Identify all properties of source figure
2. Identify all properties of target figure
3. Look for numerical correspondences
4. Check property-to-property mappings
5. Formulate relationship rule
6. Verify on all given pairs
7. Apply to predict unknown

COMMON MISTAKES TO AVOID:
- Looking only for visual transformations
- Missing abstract numerical relationships
- Not considering shape properties (corners, sides, etc.)
- Assuming relationship must be geometric
- Not verifying hypothesis on all pairs

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: Track the position of the dot in each figure
- Figure 1: Position at (40, 60)
- Figure 2: Position at (60, 40)
- Figure 3: Position at (80, 60)
- Figure 4: Position at (60, 80)

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 spiral inward pattern

RULE HYPOTHESIS:
Systematic positional movement following spiral inward pattern

VERIFICATION:
All movements conform to the identified pattern ✓

APPLICATION:
Next position: (50, 60)
Following the established spiral inward pattern

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

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

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: 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

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

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

RULE HYPOTHESIS:
The shading pattern cycles through black → gray → white

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

HIERARCHICAL RULE DETECTION:

LEVEL 1 ANALYSIS - Outer Shape Pattern:
Step 1: Identify shapes by counting sides
- Figure 1: Triangle (3 sides)
- Figure 2: Square (4 sides)
- Figure 3: Pentagon (5 sides)
- Figure 4: Hexagon (6 sides)

Rule 1 (Outer): Shape gains +1 side per step

LEVEL 2 ANALYSIS - Inner Element Pattern:
Step 2: Count dots inside each shape
- Figure 1: 1 dot
- Figure 2: 2 dots
- Figure 3: 3 dots
- Figure 4: 4 dots

Rule 2 (Inner): Number of dots increases by +1 per step

NESTED RULE HYPOTHESIS:
TWO independent hierarchical rules:
1. Outer rule: Polygon sides = 3, 4, 5, 6, ...
2. Inner rule: Dot count = 1, 2, 3, 4, ...

CORRELATION CHECK:
Both patterns follow same arithmetic progression (n+1)
Correlation: dot count = shape sides - 2 ✓

VERIFICATION:
- Figure 1: 3 sides, 1 dot (3-2=1) ✓
- Figure 2: 4 sides, 2 dots (4-2=2) ✓
- Figure 3: 5 sides, 3 dots (5-2=3) ✓
- Figure 4: 6 sides, 4 dots (6-2=4) ✓

APPLICATION:
Figure 5 should have:
- Outer: 6 + 1 = 7 sides (heptagon)
- Inner: 4 + 1 = 5 dots
- Verification: 7 - 2 = 5 ✓

ADVANCED NESTED RULE TECHNIQUES:
- Analyze each level independently
- Check for correlations between levels
- Verify consistency across all levels
- Apply all rules to predict next state
- Cross-verify using relationships

COMMON MISTAKES TO AVOID:
- Analyzing only one level
- Missing the relationship between levels
- Not verifying correlation formulas
- Applying only one rule to prediction
- Incomplete hierarchical decomposition

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: Identify each shape by counting sides
- Figure 1: 5 sides (pentagon)
- Figure 2: 6 sides (hexagon)
- Figure 3: 7 sides (heptagon)
- Figure 4: 8 sides (octagon)

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 gains one side in sequential transformation

VERIFICATION:
Consistent transformation: +1 side per step ✓

APPLICATION:
Figure 5: 8 + 1 = 9 sides
Shape: 9-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: Count elements in each figure
- Figure 1: 6 squares
- Figure 2: 5 squares
- Figure 3: 4 squares
- Figure 4: 3 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 decreasing by 1 in each step

VERIFICATION:
All consecutive differences are consistent: -1 ✓

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

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

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: 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

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: Track the position of the dot in each figure
- Figure 1: Position at (40, 60)
- Figure 2: Position at (60, 40)
- Figure 3: Position at (80, 60)
- Figure 4: Position at (60, 80)

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 spiral inward pattern

RULE HYPOTHESIS:
Systematic positional movement following spiral inward pattern

VERIFICATION:
All movements conform to the identified pattern ✓

APPLICATION:
Next position: (50, 60)
Following the established spiral inward pattern

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

PATTERN ANALYSIS:
Step 1: Examine Figure 1 and Figure 2 - note the orientation change
Step 2: Check if rotation: No consistent rotation angle found
Step 3: Check for reflection: Yes! Figure 2 is a mirror image of Figure 1
Step 4: Identify reflection axis: vertical axis
Step 5: Verify pattern: Figure 3 is reflection of Figure 2 (back to original orientation)

RULE HYPOTHESIS:
The figures alternate between original and reflected across vertical axis

VERIFICATION:
- Figure 1: Original position
- Figure 2: Reflected across vertical axis ✓
- Figure 3: Reflected back to original ✓

APPLICATION:
Figure 4 should be reflected version (same as Figure 2)
Pattern: Original → Reflected → Original → Reflected

COMMON MISTAKES TO AVOID:
- Confusing reflection with rotation
- Identifying wrong axis of reflection
- Not recognizing alternating pattern

PATTERN ANALYSIS:
Step 1: Examine Figure 1 and Figure 2 - note the orientation change
Step 2: Check if rotation: No consistent rotation angle found
Step 3: Check for reflection: Yes! Figure 2 is a mirror image of Figure 1
Step 4: Identify reflection axis: horizontal axis
Step 5: Verify pattern: Figure 3 is reflection of Figure 2 (back to original orientation)

RULE HYPOTHESIS:
The figures alternate between original and reflected across horizontal axis

VERIFICATION:
- Figure 1: Original position
- Figure 2: Reflected across horizontal axis ✓
- Figure 3: Reflected back to original ✓

APPLICATION:
Figure 4 should be reflected version (same as Figure 2)
Pattern: Original → Reflected → Original → Reflected

COMMON MISTAKES TO AVOID:
- Confusing reflection with rotation
- Identifying wrong axis of reflection
- Not recognizing alternating pattern