Rule Detection - Intermediate-Advanced Level: visual criteria INTERMEDIATE-ADVANCED

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

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: Union (A ∪ B)

RULE HYPOTHESIS:
The rule is a Union (A ∪ B) operation

SET OPERATION DEFINITION:
Union (A ∪ B) combines all elements from both sets (no duplicates)

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

Union (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: Compare consecutive figures - observe the orientation changes
Step 2: Measure rotation angle between Figure 1 and Figure 2 = 90°
Step 3: Verify this pattern: Figure 2→3 also shows 90° rotation
Step 4: Verify Figure 3→4 also follows the same 90° rotation

RULE HYPOTHESIS:
The figure rotates clockwise by 90° in each step

VERIFICATION:
- Figure 1 to 2: 90° clockwise ✓
- Figure 2 to 3: 90° clockwise ✓
- Figure 3 to 4: 90° clockwise ✓

APPLICATION:
Figure 5 should be Figure 4 rotated clockwise by 90°
Total rotation from start = 360°

COMMON MISTAKES TO AVOID:
- Confusing clockwise with counterclockwise rotation
- Measuring rotation incorrectly
- Assuming different rotation angles

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

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 triangles is increasing by 1 in each step

VERIFICATION:
All consecutive differences are consistent: +1 ✓

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

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

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

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: Count elements in each figure
- Figure 1: 2 triangles
- Figure 2: 3 triangles
- Figure 3: 4 triangles
- Figure 4: 5 triangles

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 triangles is increasing by 1 in each step

VERIFICATION:
All consecutive differences are consistent: +1 ✓

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

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

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

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: 7 + 1 = 8 sides
Shape: octagon

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 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

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

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: Compare consecutive figures - observe the orientation changes
Step 2: Measure rotation angle between Figure 1 and Figure 2 = 90°
Step 3: Verify this pattern: Figure 2→3 also shows 90° rotation
Step 4: Verify Figure 3→4 also follows the same 90° rotation

RULE HYPOTHESIS:
The figure rotates clockwise by 90° in each step

VERIFICATION:
- Figure 1 to 2: 90° clockwise ✓
- Figure 2 to 3: 90° clockwise ✓
- Figure 3 to 4: 90° clockwise ✓

APPLICATION:
Figure 5 should be Figure 4 rotated clockwise by 90°
Total rotation from start = 360°

COMMON MISTAKES TO AVOID:
- Confusing clockwise with counterclockwise rotation
- Measuring rotation incorrectly
- Assuming different rotation angles

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

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

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:

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

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: Identify each shape by counting sides
- Figure 1: 5 sides (pentagon)
- Figure 2: 4 sides (quadrilateral)
- Figure 3: 3 sides (triangle)
- Figure 4: 2 sides (2-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: 2 - 1 = 1 sides
Shape: 1-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

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