Analogy

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Subtopic 5 of 6·← Number analogy·Mixed clusters and same-relation selection →

Number sets (triads) analogy

Two or more sets of numbers such as (6, 13, 27) and (8, 17, 35) share one hidden rule; you pick the option set that follows the same rule. The rule links all numbers of a set — a repeated step (×2 + 1), the third made from the first two (a × b, a² + b²), powers (a, a², a³) or a middle number made from the outer two. Most papers add the note that operations must be done on whole numbers, so digit tricks are not allowed.

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 12 practice Q

Detailed notes

What is a number-set analogy?

This is a newer and very common form. You are shown two or more sets of numbers, such as (6, 13, 27) and (8, 17, 35). All numbers inside a set are linked by the same rule. You must pick the option set whose numbers follow the same rule.

The question usually carries a note: "Operations should be performed on the whole numbers, without breaking down the numbers into their constituent digits. E.g. 13: operations on 13 such as adding, subtracting or multiplying 13 are allowed; breaking 13 into 1 and 3 is not allowed." So only use +, −, ×, ÷, squares and cubes of the full numbers. No digit tricks.

How the rule can be built

There are four families you will meet again and again.

1. Chain rule (first → second → third). The same step is used twice. In (6, 13, 27): 6 × 2 + 1 = 13 and 13 × 2 + 1 = 27. Check (8, 17, 35): 8 × 2 + 1 = 17, 17 × 2 + 1 = 35 ✓. The step can also change: (4, 9, 27) is +5 then × 3.

2. Third from the first two. The third number is made from the other two: product (3, 8, 24), sum of squares (4, 5, 41 because 16 + 25 = 41), twice the sum (5, 9, 28).

3. Power sets. Numbers are powers or near-powers of the first: (2, 4, 8) is a, a², a³; (3, 10, 28) is a, a² + 1, a³ + 1; (7, 49, 56) is a, a², a + a².

4. Middle links the outer two. The middle number is made from the two outer numbers: (5, 35, 7) because 5 × 7 = 35; (8, 13, 18) because 13 is the average of 8 and 18. Or the first is the sum of the other two: (14, 5, 9).

Method

Take the first model set. Try the chain rule: how does the 1st become the 2nd? Does the same step turn the 2nd into the 3rd?
If not, try the third from the first two: a + b, a × b, a² + b², (a + b) × k.
If not, try powers, then the middle-link family.
Confirm the rule on the second model set. A rule that fits only one set is wrong.
Test the four options one by one. Exactly one fits. Traps are usually off by 1 to 3 in the last number.

Worked example

Sets: (4, 5, 41), (3, 7, 58). Chain? 4 → 5 is +1 but 5 → 41 is not +1. Third from first two: 4² + 5² = 16 + 25 = 41 ✓; 3² + 7² = 9 + 49 = 58 ✓. Options: (2, 6, 40), (5, 4, 40), (6, 3, 36), (1, 8, 64). 2² + 6² = 4 + 36 = 40 ✓; 25 + 16 = 41 ✗; 36 + 9 = 45 ✗; 1 + 64 = 65 ✗. Answer: (2, 6, 40).

Speed tips

Check the last number first in each option — traps usually change only that.
If the numbers grow about ×2 each step, try ×2 ± 1; about ×3, try ×3 ± 1.
Numbers like 26, 37, 50, 65 are squares + 1; 28, 65, 126 are cubes + 1.

Quick revision

Obey the whole-number note: no digit sums or products.
Four families: chain (a → b → c), third from first two, power sets, middle links outer.
The rule must fit every model set before you test options.
Test all options fully; exactly one follows the rule.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Chain rule sets (a → b → c by the same step)very common3 practice Q

How to spot it:

In each set the numbers grow step by step: (6, 13, 27), (8, 17, 35).

b = k a + r,\; c = k b + r

Find the step from the 1st to the 2nd number (× k ± r or + r).
Check the same step takes the 2nd to the 3rd.
Confirm on the other model set, then test the options.

Why it works: a repeated step is the simplest rule that links three numbers.

Example: Sets (6, 13, 27) and (8, 17, 35). Which set follows the same rule: (5, 11, 23) / (7, 15, 30) / (9, 19, 38) / (4, 9, 20)?

Rule: × 2 + 1 each step (6 → 13 → 27). (5, 11, 23): 5×2+1 = 11, 11×2+1 = 23 ✓. The others fail at the last number. Answer: (5, 11, 23).

Type 2: Third number from the first twocommon3 practice Q

How to spot it:

The third number is bigger than both and looks like a product, a sum of squares or a multiple of the sum.

c = a \times b,\; a^2+b^2,\; k(a+b)

Try a × b, a + b, a² + b², (a + b) × k on the model sets.
Keep the one that fits every model set.
Apply it to each option's first two numbers and compare with its third.

Why it works: the first two numbers are 'inputs' and the third is the 'output'.

Example: Sets (3, 8, 24) and (5, 6, 30). Which follows the same rule: (7, 4, 28) / (6, 5, 32) / (9, 3, 36) / (8, 2, 18)?

Rule: third = first × second (3 × 8 = 24, 5 × 6 = 30). 7 × 4 = 28 ✓; 30, 27, 16 fail the others. Answer: (7, 4, 28).

Type 3: Power sets (a, a², a³ and near-powers)common3 practice Q

How to spot it:

The second number is the square of the first, the third looks like a cube, or both are 'square + 1' and 'cube + 1'.

(a,\; a^2+1,\; a^3+1)

Compare the 2nd number with a² and the 3rd with a³ (or a + a²).
Note any ± r.
Test each option by building a², a³ from its first number.

Why it works: powers grow so fast that they are easy to recognise.

Example: Sets (3, 10, 28) and (4, 17, 65). Which follows the same rule: (5, 26, 126) / (6, 37, 215) / (2, 5, 8) / (7, 50, 342)?

Rule: (a, a² + 1, a³ + 1). 5: 25 + 1 = 26, 125 + 1 = 126 ✓. 6 needs 217, 2 needs 9, 7 needs 344. Answer: (5, 26, 126).

Type 4: Middle number links the outer two (or first = sum of the others)occasional3 practice Q

How to spot it:

The middle number is the product or the average of the outer two, or the first number equals the sum of the other two.

b = a \times c,\; b = \tfrac{a+c}{2}

Multiply, add and average the outer numbers; compare with the middle.
Also test first = second + third.
Confirm on all model sets, then test the options.

Why it works: setters sometimes hide the 'output' in the middle instead of at the end.

Example: Sets (5, 35, 7) and (4, 36, 9). Which follows the same rule: (6, 48, 8) / (3, 24, 9) / (7, 56, 9) / (2, 18, 8)?

Middle = first × third (5 × 7 = 35, 4 × 9 = 36). 6 × 8 = 48 ✓; 27, 63 and 16 fail the others. Answer: (6, 48, 8).

Formulas

Chain rule

b = ka + r,\; c = kb + r

Same step applied twice.

Third from first two

c = ab,\; a^2 + b^2,\; k(a+b)

Test on every model set.

Power set

(a,\; a^2 \pm r,\; a^3 \pm r)

Near-squares and near-cubes.

Shortcut tricks

⚡ Check the last number first

Build the rule from the model sets, then for each option compute only what the last number should be. Traps nearly always change the last number by 1 to 3, so this finds the answer in one pass.

Example: Q. Sets (2, 4, 8) and (3, 9, 27). Which follows the same rule: (4, 16, 60) / (5, 25, 125) / (6, 36, 206) / (7, 49, 334)?

Rule (a, a², a³). Last numbers should be 64, 125, 216, 343. Only (5, 25, 125) matches. Answer: (5, 25, 125).

Where students lose marks

Breaking numbers into digits when the question says to use whole numbers only.
Accepting a rule that fits the first model set but not the second.
Checking only the first two numbers of an option and missing that the third is off by one.

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

Suggested time 10 min · wrong answers go to your mistake notebook automatically.