Analogy

high importance~3 Q in Tier 112 formulas⚡ 11 shortcuts6 subtopics

Overview Learn (6)Formulas (12)Examples (4)Practice (83)

Number sets (triads) analogy

⏱ 5 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

You are shown two or more sets of three numbers, such as (6, 13, 27) and (8, 17, 35). One hidden rule links the numbers inside every set. Find that rule, check it on every set, then pick the option set that follows it. Most papers say to use whole numbers only, so digit tricks are not allowed.

What the question looks like

You see sets like (6, 13, 27) and (8, 17, 35). The same rule links the three numbers in each set. You pick the option set that follows the same rule.

The question usually has a note. It says to work on whole numbers only. For 13, you may add, subtract or multiply 13. You may not break it into 1 and 3.

Rule: With the whole-number note, use only +, −, ×, ÷, squares and cubes of full numbers. No digit sums.

Family one: a chain

The same step turns the 1st number into the 2nd, and the 2nd into the 3rd.

(6, 13, 27): 6 × 2 + 1 = 13, and 13 × 2 + 1 = 27. Check (8, 17, 35): 8 × 2 + 1 = 17, and 17 × 2 + 1 = 35.

The two steps can also differ: (4, 9, 27) is + 5, then × 3.

Family two: the third from the first two

The first two numbers are the inputs. The third is the output.

Product: (3, 8, 24), since 3 × 8 = 24.
Sum of squares: (4, 5, 41), since 16 + 25 = 41.
Twice the sum: (5, 9, 28), since 2 × 14 = 28.
Difference times k: (15, 9, 24), since (15 − 9) × 4 = 24.

Family three: powers

The numbers are powers of the first, or close to them.

(2, 4, 8) is a, a², a³.
(3, 10, 28) is a, a² + 1, a³ + 1.
(7, 49, 56) is a, a², a + a².

Tip: 26, 37, 50, 65 are squares + 1. 28, 65, 126 are cubes + 1. Spot them on sight.

Family four: the middle links the outer two

(5, 35, 7): the middle is the product, 5 × 7 = 35.
(8, 13, 18): the middle is the average of 8 and 18.
(14, 5, 9): the first is the sum of the other two.

Method

Take the first model set. Try a chain: does one step work twice?
If not, try the third from the first two: sum, product, sum of squares, difference × k.
If not, try powers, then the middle-link family.
Check the rule on the second model set. A rule that fits one set only is wrong.
Test the options. Exactly one fits.

Example: Sets (4, 5, 41) and (3, 7, 58). A chain fails. Sum of squares works: 16 + 25 = 41 and 9 + 49 = 58. Options: (2, 6, 40), (5, 4, 40), (6, 3, 36), (1, 8, 64). Only 4 + 36 = 40 fits. The answer is (2, 6, 40).

Watch: Traps usually change only the last number, by 1 to 3. Compute the last number for every option.

Speed checks

Check the last number of each option first. Traps usually change only that one.
Numbers growing about twice each step: try × 2, then ± 1. Growing about three times: try × 3, then ± 1.
Learn the near-power numbers on sight. 26, 37, 50 and 65 are squares + 1. 28, 65 and 126 are cubes + 1.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Chain sets (the same step twice)

How to spot it:

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

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

Method

Find the step from the 1st to the 2nd number.
Check that the same step takes the 2nd to the 3rd.
Confirm on the other model set.
Test each option.

Why it works:

A repeated step is the simplest rule that links three numbers.

Try this

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

Show solution

Rule: × 2, then + 1, at each step.
(a): $5 \times 2 + 1 = 11$ and $11 \times 2 + 1 = 23$ . It fits.
The others need 31, 39 and 19 as the last number.

Answer

(a) (5, 11, 23)

Type 2common3 practice Q

Third number from the first two

How to spot it:

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

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

Method

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

Why it works:

The first two numbers act as inputs and the third as the output.

Try this

Sets (3, 8, 24) and (5, 6, 30). Which set follows the same rule? (a) (7, 4, 28) (b) (6, 5, 32) (c) (9, 3, 36) (d) (8, 2, 18)

Show solution

Rule: third = first × second. $3 \times 8 = 24$ , $5 \times 6 = 30$ .
(a): $7 \times 4 = 28$ . It fits.
The others give 30, 27 and 16, not their third numbers.

Answer

(a) (7, 4, 28)

Type 3common3 practice Q

Power sets (a, a², a³ and near-powers)

How to spot it:

The second number is the square of the first, and the third looks like a cube (sometimes each + 1).

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

Method

Compare the 2nd number with a² and the 3rd with a³.
Note any ± r.
Build a² and a³ from each option's first number and compare.

Why it works:

Powers grow so fast that they are easy to recognise.

Try this

Sets (3, 10, 28) and (4, 17, 65). Which set follows the same rule? (a) (5, 26, 126) (b) (6, 37, 215) (c) (2, 5, 8) (d) (7, 50, 342)

Show solution

Rule: $a,\ a^2 + 1,\ a^3 + 1$ .
(a): $25 + 1 = 26$ and $125 + 1 = 126$ . It fits.
6 needs 217, 2 needs 9 and 7 needs 344.

Answer

(a) (5, 26, 126)

Type 4occasional3 practice Q

Middle number links the outer two

How to spot it:

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

b = a \times c,\ b = \dfrac{a+c}{2}

Method

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.

Try this

Sets (5, 35, 7) and (4, 36, 9). Which set follows the same rule? (a) (6, 48, 8) (b) (3, 24, 9) (c) (7, 56, 9) (d) (2, 18, 8)

Show solution

Rule: middle = first × third. $5 \times 7 = 35$ , $4 \times 9 = 36$ .
(a): $6 \times 8 = 48$ . It fits.
The others give 27, 63 and 16.

Answer

(a) (6, 48, 8)

Type 5occasional

Third number from the difference

How to spot it:

The third number is not close to the product or the sum, but the first two numbers are close to each other.

c = k(a - b)

Method

Find a − b for each model set.
See how a − b becomes the third number (often × k).
Check on the second model set.
Apply to each option.

Why it works:

When the sum and product are too big, the small gap between the first two is the next thing to test.

Try this

Sets (15, 9, 24) and (20, 13, 28). Which set follows the same rule? (a) (18, 11, 28) (b) (16, 10, 20) (c) (25, 19, 20) (d) (14, 6, 36)

Show solution

$(15 - 9) \times 4 = 24$ and $(20 - 13) \times 4 = 28$ . Rule: difference × 4.
(a): $(18 - 11) \times 4 = 28$ . It fits.
The others give 24, 24 and 32.

Answer

(a) (18, 11, 28)

Formula sheet

Chain rule

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

a, b, c are the three numbers of a set. One step is used twice.

Third from first two

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

Test on every model set.

Power set

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

Near-squares and near-cubes of the first number.

Shortcuts that save time

⚡ Compute only the last number

Once you have the rule, work out what the last number of each option should be. Traps nearly always change only that number, so one pass finds the answer.

Example

Sets (2, 4, 8) and (3, 9, 27). Which set follows the same rule? (a) (4, 16, 60) (b) (5, 25, 125) (c) (6, 36, 206) (d) (7, 49, 334)

Show solution

Rule: a, a², a³.
The last numbers should be 64, 125, 216 and 343.
Only option (b) has the right last number.

Answer

(b) (5, 25, 125)

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Breaking numbers into digits when the whole-number note is given.

Use only operations on full numbers: +, −, ×, ÷, squares, cubes.

Mistake 02

Accepting a rule that fits the first model set but not the second.

Check the rule on every model set before looking at the options.

Mistake 03

Checking only the first two numbers of an option.

Compute the third number too. Traps are usually off by 1 to 3 there.

Mistake 04

Trying only chains and giving up.

Go through the four families in order: chain, third from two, powers, middle link.

Quick revision

Read this the night before the exam.

Whole-number note: no digit tricks.
Four families: chain, third from the first two, powers, middle links the outer two.
The rule must fit every model set.
Compute the last number of every option; traps change only that.
Squares + 1: 10, 17, 26, 37, 50, 65. Cubes + 1: 9, 28, 65, 126.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

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

Continue learning