Classification (Odd One Out)

high importance~3 Q in Tier 19 formulas⚡ 9 shortcuts6 subtopics

Overview Learn (6)Formulas (9)Examples (4)Practice (89)

Number pairs and sets odd one out

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

The idea in one minute

Here each option is not one number but a small group: a pair like 12 : 144 or a set like (3, 9, 27). Three options are built by the same operation and one is not. Find the operation from one option, confirm it on a second, then mark the option that fails. Many papers add a note: use whole numbers only, so digit tricks are not allowed.

Pairs and sets instead of single numbers

Each option is a pair like 12 : 144, or a set of three numbers like (3, 9, 27). Three options follow the same operation. One does not. You hunt for the operation that links the numbers inside each option.

Take (2, 4, 8), (3, 6, 12), (5, 10, 20), (4, 8, 15). In the first three sets each number is double the one before. In the last set 8 × 2 = 16, not 15. So (4, 8, 15) is odd.

Rule: Build the rule from one option, confirm it on a second, then mark the option that breaks it.

The whole-number note

Many papers print a note: "operations should be performed on whole numbers, without breaking them into digits". So 13 must stay 13. You may not split it into 1 and 3. Digit sums and digit products are not allowed in these questions.

Pairs: the second number from the first

The second number is built from the first. Common rules:

$b = a^2$ , as in 12 : 144. Or $b = a^3$ .
$b = a^2 + k$ , as in 4 : 19, which is $4^2 + 3$ .
$b = a^3 + 1$ , as in 5 : 126.
$b = k a + r$ , as in 7 : 22, which is 3 × 7 + 1, or 3 : 7, which is 2 × 3 + 1.

Chain sets: one step, used twice

The same step turns the first number into the second, and the second into the third.

(2, 7, 17): 2 × 2 + 3 = 7 and 7 × 2 + 3 = 17. The rule is "×2 + 3". Check both steps of every set, never only the first.

The middle from the outer two

The middle number is built from the first and last numbers:

$m = a \times c$ , as in (4, 20, 5), because 4 × 5 = 20.
$m = a^2 + c^2$ , as in (5, 34, 3), because 25 + 9 = 34.
$m = (a + c) \times k$ .

The third from the first two

The last number is built from the first two: $c = 2(a + b)$ as in (12, 8, 40), or $c = a \times b - 1$ as in (7, 3, 20).

Tip: Compare sizes to pick the family. A large middle number points to a product. Steady growth points to a chain. A large last number points to a sum or product rule.

Select the set that belongs

Some questions give two model sets, like (3, 12, 48) and (2, 8, 32), both built by ×4 twice. You pick the option that follows the same rule. Exactly one option fits; the others break it at the second step.

Worked example

Odd set: (6, 11, 21), (8, 15, 29), (5, 9, 17), (7, 13, 26).

6 × 2 − 1 = 11 and 11 × 2 − 1 = 21. Rule: ×2 − 1 twice.
8 → 15 → 29 fits. 5 → 9 → 17 fits.
7 × 2 − 1 = 13, but 13 × 2 − 1 = 25, not 26.

Answer: (7, 13, 26).

Watch: The setter usually spoils only the last number of one option. Always finish the chain.

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

Number pairs with one wrong relation (a : b)

How to spot it:

Options are pairs like 12 : 144 or 4 : 19. The second number is a square, cube or simple function of the first.

b = a^2 \pm k \;\text{or}\; b = a^3 \pm k \;\text{or}\; b = ka \pm r

Method

Guess the rule from the first pair (square? cube? square + k?).
Confirm it on a second pair.
The pair that fails is odd.

Why it works:

One pair is built with the wrong constant or the wrong power, so a full check catches it.

Try this

Find the odd pair: 12 : 144, 15 : 225, 9 : 81, 11 : 111.

Show solution

144 = 12², 225 = 15², 81 = 9². Rule: second = first squared.
11² = 121, not 111.
Rule: three square pairs, one broken pair.

Answer

11 : 111

Type 2common3 practice Q

Chain sets (a, b, c): the same operation twice

How to spot it:

Sets of three numbers that grow steadily, like (3, 9, 27) or (2, 7, 17).

(a,\ f(a),\ f(f(a)))

Method

Find the operation from a to b.
Check that the same operation turns b into c.
The set where the second step fails is odd.

Why it works:

The setter usually spoils only the last number of one set.

Try this

Find the odd set: (3, 9, 27), (4, 16, 64), (5, 25, 125), (6, 36, 206).

Show solution

Each set is (n, n², n³): 3³ = 27, 4³ = 64, 5³ = 125.
6³ = 216, but the set says 206.
Rule: three n, n², n³ sets, one broken at the cube.

Answer

(6, 36, 206)

Type 3common2 practice Q

Middle number from the two outer numbers

How to spot it:

Sets of three where the middle number is much larger than the two outer numbers.

m = a \times c \;\text{or}\; m = a^2 + c^2

Method

Try middle = first × last.
If not, try first² + last², or (first + last) × k.
The set that fails is odd.

Why it works:

The outer numbers are the inputs and the middle is the output.

Try this

Find the odd set: (4, 20, 5), (3, 21, 7), (6, 48, 8), (9, 45, 6).

Show solution

4 × 5 = 20, 3 × 7 = 21, 6 × 8 = 48. Rule: middle = first × last.
9 × 6 = 54, but the set says 45.
Rule: three product sets, one broken.

Answer

(9, 45, 6)

Type 4occasional2 practice Q

Third number from the first two

How to spot it:

Sets where the last number is the largest and looks like a sum or product of the first two.

c = k(a + b) \;\text{or}\; c = ab \pm r

Method

Try c = a + b, c = k(a + b), then c = a × b ± r.
Confirm the rule on two sets.
The set that breaks it is odd.

Why it works:

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

Try this

Find the odd set: (12, 8, 40), (15, 5, 40), (9, 7, 32), (10, 6, 30).

Show solution

12 + 8 = 20 and 2 × 20 = 40. 15 + 5 = 20 and 2 × 20 = 40. 9 + 7 = 16 and 2 × 16 = 32. Rule: c = 2(a + b).
10 + 6 = 16, so the third number should be 32, not 30.
Rule: three sum-double sets, one broken.

Answer

(10, 6, 30)

Type 5common2 practice Q

Select the set or pair that belongs to the group

How to spot it:

The question gives two sets or pairs as a model and asks which option is related in the same way.

Method

Find the rule that fits BOTH model sets.
Test every option against it.
Exactly one option fits every step — choose it.

Why it works:

This is classification turned around: three options break the rule and one follows it.

Try this

Which set is like (3, 12, 48) and (2, 8, 32)? (5, 20, 80), (4, 12, 36), (6, 24, 72), (7, 28, 84).

Show solution

Model rule: ×4, then ×4 again. 3 × 4 = 12, 12 × 4 = 48; 2 × 4 = 8, 8 × 4 = 32.
(5, 20, 80): 5 × 4 = 20 and 20 × 4 = 80. It fits both steps.
(6, 24, 72) and (7, 28, 84) break the second step; (4, 12, 36) breaks the first.

Answer

(5, 20, 80)

Formula sheet

Pair rule

b = f(a),\; f(a) \in \{a^2,\ a^3,\ a^2 \pm k,\ ka \pm r\}

Find f from one pair, then check the other three pairs.

Chain triad

(a,\ f(a),\ f(f(a)))

The same operation applied twice, e.g. ×3 gives (3, 9, 27).

Outer-to-middle triad

(a,\ g(a, c),\ c),\; g = ac \text{ or } a^2 + c^2

The middle number is made from the two outer numbers.

First-two-to-third triad

(a,\ b,\ h(a, b)),\; h = k(a+b) \text{ or } ab \pm r

The third number is made from the first two.

Shortcuts that save time

⚡ Size tells the operation

Compare sizes first. If the second number is near the square of the first, think square. If the middle number is large and the outer ones small, think product of the outer numbers.

Example

Find the odd set: (4, 20, 5), (3, 21, 7), (6, 48, 8), (9, 45, 6).

Show solution

The middle numbers are large; the outer numbers are small. Try middle = first × last.
4 × 5 = 20, 3 × 7 = 21, 6 × 8 = 48. All fit.
9 × 6 = 54, but the set says 45.

Answer

(9, 45, 6)

⚡ Test the rule on two options before trusting it

A rule found from one option can be a coincidence. Confirm it on a second option, then check the rest. The option that fails is the answer.

Example

Find the odd pair: 4 : 19, 6 : 39, 7 : 52, 8 : 66.

Show solution

4 : 19 could be 4 × 4 + 3 or 4 × 5 − 1.
Test on 6 : 39: 6² + 3 = 39 fits, while 6 × 7 − 1 = 41 does not. Rule: a² + 3.
7² + 3 = 52 fits. But 8² + 3 = 67, not 66.

Answer

8 : 66

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Splitting numbers into digits (12 read as 1 and 2) when the whole-number note is printed.

Use only +, −, ×, ÷ and powers of the full numbers.

Mistake 02

Fixing the rule from only one option.

One option fits many rules. Confirm on a second option before testing the rest.

Mistake 03

Choosing the option with the largest numbers.

Mark the option that breaks the rule, whatever its size.

Mistake 04

In "select the set like the given sets", picking a set that looks similar but breaks the rule.

Test every option against the model rule. Exactly one fits all steps.

Mistake 05

Stopping after the first step of a chain.

Check both steps of every set; the trap usually hides in the last number.

Quick revision

Read this the night before the exam.

Whole-number note: no digit tricks.
Pairs: try a², a³, a² ± k, ka ± r.
Chain sets: one step used twice; check both steps.
Middle from outer: a × c, a² + c², k(a + c).
Third from first two: k(a + b), ab ± r.
Belongs-to-group: find the rule of the model sets, then pick the one fit.

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 8 min · wrong answers go to your mistake notebook automatically.

Continue learning