SPI Reasoning · Classification

Odd One Out

Three of the four options share a rule and one breaks it. Find the rule from the options, test it on each, and the one that fails is the answer.

All Classification notes

Why this matters

Six questions in the bank, all from the 2022 paper: three EASY, one MODERATE and two HARD. Three ask for the odd number and three for the odd pair or set, and every one is settled by a number fact or a rule tested on each option in turn.

Concept 1 of 3: Find the rule three options share

You are not looking for the strange option first. You are looking for what three of them have in common. Once you see the shared rule, the odd one shows itself.

Definition

The method:

Try the quick rules first: even or odd, multiples of one number, primes, squares, cubes, sum of digits.
A rule counts only if exactly three options keep it and one breaks it.
If two different rules each pick a different option, prefer the rule that explains the options as written, such as why one number has more digits.
Test the rule on every option, not just the first two.

Worked example

Find the odd number: (a) 14 (b) 21 (c) 35 (d) 40

Practice this conceptself-check · 4 quick reps

Stopping at the first rule you see

A rule that two options keep and two break picks out nothing. Keep trying until three options keep it and one breaks it.

Testing only some options

An option that looks like the others can still break the rule. Check every option before you mark.

Concept 2 of 3: Number facts to test first

Most odd-number questions hide a plain fact: three primes and one composite, three squares and one near miss. Know these lists well and you can check an option in seconds.

Definition

Keep these in mind:

2 is the only even prime. Any other even number is not prime.
To test a number for prime, divide by the primes up to its square root.
A perfect square lies exactly on a square; find the two squares a number sits between.
A number is divisible by 3 (or 9) when its digit sum is.

Property	Numbers to know	Quick test
Primes up to 100	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97	Divide by 2, 3, 5, 7 for numbers under 100 51 = 3 × 17, 57 = 3 × 19 and 91 = 7 × 13 look prime but are not.
Perfect squares	121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 625, 900, 1024	Find the squares on either side
Perfect cubes	1, 8, 27, 64, 125, 216, 343, 512, 729, 1000	Learn them to 10 cubed
Even numbers	Last digit 0, 2, 4, 6 or 8	Look at the last digit
Digit sum	Add the digits: 5742 gives 18	Same digit sum is a common shared rule

Try the quick tests in this order: even or odd, prime, square or cube, digit sum.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q47Easy

Example 2 · Classification · Odd One Out

In the following five questions find out odd number or odd set of numbers from the given options.

Calling an odd number prime on sight

Odd is not the same as prime. Test small divisors: 87 is

3 \times 29

and 91 is

7 \times 13

Missing a second rule

When one number is the only even one and another has a different digit sum, two rules point two ways. Check which rule explains all four options before you choose.

Drill 2 more on number facts to test first

Concept 3 of 3: The rule inside a pair or set

When each option is a pair or a set, the rule links the numbers inside it. Find how the first number turns into the others in one option, then test the same rule on the rest.

Definition

Rules to try, in this order:

Second is the square or cube of the first, or one more or less than it.
Last is the sum, difference or product of the others.
Last is the first divided by the second.
Two of the numbers are consecutive.
The product's digits are rearranged to make the last number.

Work the rule out on the simplest option, then check it on all four.

Common links in a set

b = a^2,\; b = a^3,\; b = a^2 \pm 1,\; c = a \times b,\; c = a \div b

Worked example

Find the odd pair: (a) 4, 17 (b) 6, 37 (c) 3, 10 (d) 5, 24

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q49Hard

Example 3 · Classification · Odd One Out

In the following five questions find out odd number or odd set of numbers from the given options.

Finding a rule that fits two options only

A rule found on one option must be checked on all four. If it fails on two, it is the wrong rule, so try the next one on the list.

Missing a near miss

An odd set often differs by very little, such as a cube that is 10 too big. Work each value out in full rather than judging by size.

Drill 2 more on the rule inside a pair or set

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The rule inside a pair or set
Common links in a set
$b = a^2,\; b = a^3,\; b = a^2 \pm 1,\; c = a \times b,\; c = a \div b$

Reference tables (1)

Number facts to test first5 rows

Property	Numbers to know	Quick test
Primes up to 100	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97	Divide by 2, 3, 5, 7 for numbers under 100 51 = 3 × 17, 57 = 3 × 19 and 91 = 7 × 13 look prime but are not.
Perfect squares	121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 625, 900, 1024	Find the squares on either side
Perfect cubes	1, 8, 27, 64, 125, 216, 343, 512, 729, 1000	Learn them to 10 cubed
Even numbers	Last digit 0, 2, 4, 6 or 8	Look at the last digit
Digit sum	Add the digits: 5742 gives 18	Same digit sum is a common shared rule

Try the quick tests in this order: even or odd, prime, square or cube, digit sum.

Watch out for (6)

Stopping at the first rule you see
→ Find the rule three options share
Testing only some options
→ Find the rule three options share
Calling an odd number prime on sight
→ Number facts to test first
Missing a second rule
→ Number facts to test first
Finding a rule that fits two options only
→ The rule inside a pair or set
Missing a near miss
→ The rule inside a pair or set

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper Drill all 6 questions