Number-Based Patterns

Number-based patterns are among the most versatile and frequently tested rule types in UCAT Abstract Reasoning. They involve counting — shapes, sides, intersections, shading types, or any other quantifiable feature. Because these rules require mental arithmetic (however simple), they can slow you down if you are not practised. This lesson ensures you know exactly what to count and how to count it efficiently.

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What Can Be Counted?

Countable feature	Examples
Total number of shapes	"Every box contains exactly 4 shapes"
Number of a specific shape	"Every box contains exactly 2 triangles"
Total number of sides	"The total sides across all shapes in each box is 15"
Number of shapes with a specific fill	"There are always 3 black shapes"
Number of intersections	"Shapes overlap to create exactly 2 intersection points"
Number of right angles	"The total number of right angles across all shapes is always 8"
Number of straight edges	"Each box has exactly 10 straight edges"
Number of enclosed regions	"Overlapping shapes create exactly 3 enclosed regions"

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Category 1: Fixed Total Count

The simplest number-based rule: every box in the set contains the same total number of shapes.

Worked Example

Set A:

• Box 1: One circle, one triangle, one square, one pentagon (4 shapes)
• Box 2: Two hexagons, one diamond, one arrow (4 shapes)
• Box 3: One star, one rectangle, one circle, one triangle (4 shapes)
• Box 4: Four squares (4 shapes)
• Box 5: One pentagon, two circles, one triangle (4 shapes)
• Box 6: Three arrows, one hexagon (4 shapes)

Rule: Every box contains exactly 4 shapes.

Set B might have a different fixed count (e.g., always 3 shapes) or a completely unrelated rule.

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Category 2: Odd vs Even Count

Rather than a fixed number, the rule may be about whether a count is odd or even.

What Might Be Odd/Even?

• Total number of shapes
• Total number of sides
• Number of black shapes
• Number of curved shapes

Worked Example: Even Number of Shapes

Set A:

• Box 1: Two circles (2 shapes — even)
• Box 2: One triangle, one square, one pentagon, one hexagon (4 shapes — even)
• Box 3: Six small arrows (6 shapes — even)
• Box 4: One diamond, one circle (2 shapes — even)
• Box 5: Three triangles, one square (4 shapes — even)
• Box 6: Two pentagons, four circles (6 shapes — even)

Set B:

• Box 1: Three squares (3 shapes — odd)
• Box 2: One hexagon, one circle, one triangle, one arrow, one diamond (5 shapes — odd)
• Box 3: One rectangle (1 shape — odd)
• Box 4: Two circles, one pentagon (3 shapes — odd)
• Box 5: Three hexagons, two triangles (5 shapes — odd? No, that is 5 — odd ✓)
• Box 6: Seven small circles (7 shapes — odd)

Rule A: Even number of shapes. Rule B: Odd number of shapes.

Test shape: Three circles, two squares (5 shapes — odd) → Set B

Speed tip: You do not need to identify the exact count. Just determine odd or even, which is faster.

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Category 3: Total Side Count

Counting the total number of sides across all shapes in a box is one of the most common number-based rules in UCAT AR.

Side Count Reference

Shape	Sides
Circle	0
Triangle	3
Square / rectangle / diamond / parallelogram / trapezium	4
Pentagon	5
Hexagon	6
Heptagon	7
Octagon	8
Arrow (5-pointed)	5 or 7 (varies)
Star (5-pointed)	10
Cross	12

Worked Example: Total Sides Is Always 12

Set A:

• Box 1: Two hexagons (6 + 6 = 12 ✓)
• Box 2: Three squares (4 + 4 + 4 = 12 ✓)
• Box 3: Four triangles (3 + 3 + 3 + 3 = 12 ✓)
• Box 4: One hexagon, two triangles (6 + 3 + 3 = 12 ✓)
• Box 5: One square, one pentagon, one triangle (4 + 5 + 3 = 12 ✓)
• Box 6: Two pentagons, one circle (5 + 5 + 0 = 10)... Hmm, that is only 10.

Wait — this illustrates the importance of checking every box. Let me revise Box 6:

• Box 6: One octagon, one square (8 + 4 = 12 ✓)

Rule: Total sides in each box equals 12.

Test shape: One pentagon, one hexagon, one circle (5 + 6 + 0 = 11) → Does not equal 12 → Not Set A