Counting-Based Complex Rules

Counting-based rules appeared in the previous course as straightforward patterns — "always 4 shapes" or "total sides is even." In harder UCAT questions, counting becomes considerably more demanding. You might need to count sides across multiple shapes, compare the ratio of two different counts, or track a running total. This lesson covers the advanced counting patterns that appear in medium-to-hard AR questions.

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Advanced Counting Rule 1: Total Sides Equals a Specific Number

We encountered this briefly before, but in complex questions the total is higher and the shape combinations are less obvious.

Worked Example: Total Sides = 16

Set A:

• Box 1: Two pentagons (5+5) + one hexagon (6) = 16 ✓
• Box 2: Four squares (4+4+4+4) = 16 ✓
• Box 3: One octagon (8) + two squares (4+4) = 16 ✓
• Box 4: One hexagon (6) + one pentagon (5) + one triangle (3) + one circle (0) + one square (0? No — wait, that would be wrong)

Let me recalculate Box 4: hexagon (6) + pentagon (5) + triangle (3) + a shape with 2 sides? There is no common 2-sided shape. Let me redesign:

• Box 4: One hexagon (6) + one pentagon (5) + one pentagon (5) = 16 ✓
• Box 5: One hexagon (6) + one hexagon (6) + one square (4) = 16 ✓
• Box 6: Two triangles (3+3) + one pentagon (5) + one pentagon (5) = 16 ✓

Rule: Total sides always equals 16.

Speed technique: Start by counting the sides of the first box. If you get 16, check Box 2. If also 16, you have a strong hypothesis. Verify with Box 3 for confirmation.

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Advanced Counting Rule 2: More Curved Shapes Than Straight

This rule requires classifying each shape as curved or straight, then counting each group.

Worked Example

Set A:

• Box 1: Three circles (curved), one triangle (straight) → 3 curved > 1 straight ✓
• Box 2: Two circles, one semi-circle (all curved), one square (straight) → 3 curved > 1 straight ✓
• Box 3: Four circles (curved), two pentagons (straight) → 4 curved > 2 straight ✓
• Box 4: One circle (curved) — no straight shapes → 1 > 0 ✓
• Box 5: Two semi-circles, one circle (3 curved), one hexagon, one triangle (2 straight) → 3 > 2 ✓
• Box 6: Five circles (5 curved), three squares (3 straight) → 5 > 3 ✓

Set B:

• Box 1: One circle (1 curved), three triangles (3 straight) → 1 < 3 — more straight ✓
• Box 2: One semi-circle (1 curved), two hexagons, one pentagon (3 straight) → 1 < 3 ✓
• (and so on)

Rule A: More curved shapes than straight. Rule B: More straight shapes than curved.

Test shape: Two circles, two squares → 2 curved = 2 straight (equal, not "more") → Neither

The "equal" trap: When the rule is "more X than Y," equal counts satisfy neither "more X" nor "more Y." This test shape would be Neither.

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Advanced Counting Rule 3: Counting by Attribute and Comparing

Rules that require counting two different things and comparing them.

Common Comparison Rules

Rule	What you count	Comparison
"More black shapes than white"	Count black, count white	Black > white
"Number of triangles > number of circles"	Count specific shapes	Triangles > circles
"More shapes with 4+ sides than shapes with <4 sides"	Classify and count	4+ sides > <4 sides
"Number of large shapes = number of small shapes"	Count by size	Large = small

Worked Example: Triangles Exceed Circles

Set A:

• Box 1: Three triangles, one circle, two squares (3 triangles > 1 circle ✓)
• Box 2: Two triangles, one circle, one pentagon (2 > 1 ✓)
• Box 3: Four triangles, two circles (4 > 2 ✓)
• Box 4: One triangle, zero circles, three hexagons (1 > 0 ✓)
• Box 5: Five triangles, three circles, one diamond (5 > 3 ✓)
• Box 6: Two triangles, one circle (2 > 1 ✓)

Rule: Number of triangles always exceeds the number of circles. Other shapes are distractors.

Test shape: Two triangles, two circles, one square → 2 triangles = 2 circles → Not "exceeds" → Not Set A

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Advanced Counting Rule 4: Sum or Difference Rules

The rule involves a mathematical operation on two counts.

Worked Example: Black Shapes Minus White Shapes = 2