Multi-Rule Patterns

In the previous course, every example involved a single rule — one attribute that distinguished Set A from Set B. Real UCAT questions frequently go further: Set A requires two or more conditions to be met simultaneously. These are called multi-rule or compound patterns, and they are responsible for a disproportionate share of incorrect answers. This lesson teaches you to identify them reliably.

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Why Multi-Rule Patterns Are Harder

With a single rule, one pass through SCANS usually reveals the answer. With a compound rule, you face two challenges:

1. You may find the first rule and stop looking. The first rule explains most boxes, but one or two boxes seem to violate it. The temptation is to conclude you have the wrong rule — when in fact there is a second condition you have not identified.

2. Each individual rule may be less obvious. When two rules combine, neither alone produces a perfectly clean pattern. Both need to hold simultaneously, which can obscure each one.

Golden rule for compound patterns: If you find a rule that holds for 4 or 5 out of 6 boxes but fails for one, do not discard it. Instead, look for a second condition that explains the exception.

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How Compound Rules Are Constructed

Compound rules combine two (occasionally three) conditions using AND logic:

• "The total number of shapes is even AND all shapes are black"
• "Every box contains a circle AND the circle is always the largest shape"
• "There are exactly 3 shapes AND the total number of sides is 12"

Both conditions must be satisfied for a box to belong to the set. If a test shape satisfies one condition but not the other, it does not belong to the set.

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Worked Example 1: Number + Shading

Set A:

• Box 1: Three black triangles (3 shapes, all black)
• Box 2: One black circle, one black square, one black pentagon (3 shapes, all black)
• Box 3: Three black hexagons (3 shapes, all black)
• Box 4: One black diamond, one black rectangle, one black arrow (3 shapes, all black)
• Box 5: Three black circles (3 shapes, all black)
• Box 6: One black triangle, one black hexagon, one black square (3 shapes, all black)

Set B:

• Box 1: Four white triangles (4 shapes, all white)
• Box 2: Two white circles, one white pentagon (3 shapes, all white — wait, this has 3 shapes too!)

Hmm. Let me reconsider. If Set B also has 3 shapes sometimes, then "3 shapes" alone is not the distinguishing rule. Let me redesign:

Set B:

• Box 1: Two black circles, one white triangle (3 shapes, mixed colours)
• Box 2: Four black squares (4 shapes, all black — different count)
• Box 3: One black pentagon, one white hexagon, one grey diamond (3 shapes, mixed)
• Box 4: Two black arrows (2 shapes)
• Box 5: Three white rectangles (3 shapes, all white)
• Box 6: One black circle, one black triangle, one white square, one grey pentagon (4 shapes, mixed)

Analysis:

Set A: Every box has exactly 3 shapes AND every shape is black. Both conditions hold across all 6 boxes.

Set B: Some boxes have 3 shapes (Boxes 1, 3, 5) but not all black. Some have all-black shapes (Box 2: 4 black squares) but not exactly 3. No box satisfies both conditions simultaneously.

Test shapes:

• Three white triangles: 3 shapes ✓ but not all black ✗ → Not Set A
• Two black circles: all black ✓ but only 2 shapes ✗ → Not Set A
• Three black pentagons: 3 shapes ✓ AND all black ✓ → Set A

Key insight: Each condition alone is insufficient. You need both. This is the hallmark of a compound rule.

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Worked Example 2: Shape Type + Position

Set A:

• Box 1: A triangle in the centre, two circles on the left, one square on the right
• Box 2: A triangle in the centre, one pentagon on the left, three hexagons on the right
• Box 3: A triangle in the centre, one diamond on the left, one arrow on the right
• Box 4: A triangle in the centre, two rectangles on the left, one star on the right
• Box 5: A triangle in the centre, one hexagon on the left, two circles on the right
• Box 6: A triangle in the centre, three squares on the left, one pentagon on the right

Rule: Every box contains a triangle AND the triangle is always in the centre of the box. Other shapes vary (distractors) in type, number, and position.

A candidate might identify "every box has a triangle" (condition 1) but miss the positional requirement (condition 2). If a test shape has a triangle in the corner, it would violate condition 2 and not belong to Set A.

Test shapes:

• A triangle in the top-left corner, two circles in the centre → triangle present ✓ but not in centre ✗ → Not Set A
• A triangle in the centre, one hexagon on the left → triangle ✓ and in centre ✓ → Set A
• No triangle, one square in the centre → no triangle ✗ → Not Set A

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Worked Example 3: Shading + Side Count

Set A:

• Box 1: One black triangle (3), one white pentagon (5) — total: 8, even
• Box 2: Two black squares (4+4=8), one white circle (0) — total: 8, even. But also: black shapes = 2, white = 1
• Box 3: One black hexagon (6), one white square (4) — total: 10, even
• Box 4: One black triangle (3), one white triangle (3), one black circle (0), one white square (4) — total: 10, even
• Box 5: One black pentagon (5), one white pentagon (5), one black circle (0) — total: 10, even
• Box 6: Two black triangles (3+3=6), one white hexagon (6) — total: 12, even

Observations:

1. Total sides is always even: 8, 8, 10, 10, 10, 12 ✓
2. Every box has at least one black shape and at least one white shape ✓

Rule: Total number of sides is even AND both black and white shapes are present.

Test shape: Two black squares (4+4=8 sides, even ✓) but no white shapes (only black ✗) → Fails condition 2 → Not Set A

Test shape: One black triangle (3) and one white square (4) = 7 sides (odd ✗) → Fails condition 1 → Not Set A

Test shape: One black hexagon (6) and one white rectangle (4) = 10 (even ✓), both colours present ✓ → Set A

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Strategy: How to Identify Compound Rules

Step 1: Find the First Rule

Apply SCANS normally. You will likely identify one rule quickly. Verify it against all 6 boxes.

Step 2: Check for Exceptions

If the rule holds for all 6 boxes, it might be the complete rule — or there might be a second condition hiding. Look at Set B: do any Set B boxes also satisfy your first rule? If yes, there must be a second condition that Set B boxes violate.

Step 3: Find the Second Rule

Return to SCANS, but this time focus only on features you did not check the first time. If your first rule was about shape type (S), check Colour, Arrangement, Number, and Size for the second condition.

Step 4: Verify the Compound Rule

Both conditions must hold for every Set A box and must not both hold for any Set B box. It is acceptable for individual conditions to be met in Set B — it is the combination that matters.

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How Many Rules Should You Expect?

Difficulty level	Typical number of rules
Easy	1 rule
Medium	2 rules (compound)
Hard	2-3 rules, often with distractors
Very hard	2 rules where one is a conditional or relational rule

Most UCAT AR questions have 1-2 rules. Three simultaneous rules are rare but not unheard of. If you find yourself looking for a fourth rule, you have probably overcomplicated it.

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Common Compound Rule Combinations

Combination	Example
Number + Shading	"Exactly 4 shapes, all black"
Shape type + Size	"Contains a circle, and the circle is always the largest"
Shape type + Number	"Contains a triangle, and total number of shapes is odd"
Shading + Position	"Black shapes on the left, white shapes on the right"
Number + Side count	"Exactly 3 shapes, total sides = 12"
Size + Shading	"Large shapes are black, small shapes are white"

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Key Takeaways

• Multi-rule (compound) patterns require two or more conditions to be satisfied simultaneously
• If one rule explains 4-5 out of 6 boxes, do not discard it — look for a second condition
• Check Set B: if Set B boxes satisfy your first rule, there must be a distinguishing second condition
• Most questions have 1-2 rules. Three rules are rare.
• Apply SCANS twice: once to find the first rule, then again (focusing on unchecked categories) to find the second
• A test shape must satisfy all conditions to belong to a set — failing any one condition means it does not belong

In the next lesson, we examine conditional rules — a special type of compound pattern where one condition triggers another.