1. What this chapter covers
Finding the Odd One Out, Classification, and Spotting the Mismatched or Heterogeneous Item are closely related exam tasks. A set contains a rule shared by most entries. Your job is to find the entry that breaks it. Some classification questions reverse the request and ask you to name the common group or choose the item that does belong to a stated group. Read the final instruction before selecting an answer.
The rule may concern meaning, number properties, letter positions, word relationships, a figure's shape, or a combination of attributes. It can apply to a single item such as 37, a pair such as leaf : tree, or an ordered triple such as (2, 4, 8). Check each complete item, including the order of its components.
This chapter uses four-choice questions with exactly one intended odd entry. A good explanation names the common rule, verifies it for the other three choices, and shows precisely why the selected choice fails. Merely saying an item “looks different” does not establish an answer.
2. The basic classification method
- Read the requested output. Is the question asking for the odd item, the shared category, the mismatched relationship, or a member of a stated class?
- Identify each choice's type. Are they numbers, letters, words, pairs, figures, or mixed expressions?
- Test a simple common property. Start with a clear rule that applies to three of the four entries.
- Check all three matching entries independently. Do not stop after finding one attractive difference.
- Check the fourth entry against the same rule. A valid odd-one-out answer fails that exact rule.
- Write a one-sentence justification. For example: “Three numbers are perfect squares; 90 is not.”
A small attribute grid can help when the choices look similar. For number choices, record divisibility, parity, square/cube status, prime/composite status, and digit pattern. For figures, record sides, straight or curved boundary, symmetry, and rotation. The grid is a way to discover the rule; the final answer should state the rule plainly.
3. Same group does not mean identical objects
Classification groups objects by a chosen property. Violin, flute, and trumpet are different instruments, but all belong to the class “musical instruments.” A telescope does not. In another question, all four choices might be instruments, with one wind instrument among three string instruments. The relevant group changes with the other choices.
For words, prefer a precise common class: mammals, months, metals, parts of a book, or tools used to measure length. Broad labels such as “things” or “useful objects” include nearly everything and cannot explain a unique exception.
An item may belong to several groups. The answer must be based on a property shared by all the other options. If two equally natural properties leave different odd items, the question lacks a unique answer; look for a stated condition or a more precise option set.
4. Number classification
Start with familiar mathematical classes:
| Property | Quick check | Example |
|---|---|---|
| Even or odd | Divide by 2 | 18 is even; 19 is odd. |
| Prime or composite | Test divisors up to the square root | 29 is prime; 39 is composite because $39=3\times13$. |
| Perfect square | Find an integer whose square matches | $81=9^2$; 90 is not a perfect square. |
| Perfect cube | Find an integer whose cube matches | $64=4^3$; 100 is not a perfect cube. |
| Divisibility | Test the same divisor for every choice | 14, 21, and 28 are multiples of 7. |
| Digit relation | Compare digit sums, gaps, or reversal | In 235, the digit gaps are $+1,+2$. |
Do not use a different test for each option. For example, if the proposed rule is “multiple of 7,” divide each of the four entries by 7. A number's last digit alone is not enough to prove primality or perfect-square status.
Number pairs and triples
When each option is a pair, ask whether the first component transforms into the second by one operation. In 4:20, 6:42, 8:72, the second number is $n(n+1)$. The pair 10:100 breaks this rule because $10(11)=110$.
For triples, test the relationship between adjacent entries. In (2,4,8), the second is twice the first and the third is twice the second. A triple like (7,14,21) breaks the second doubling step. Preserve the order: an unordered set of the same three numbers might imply a different task.
5. Letter classification
For English letter positions use A = 1, B = 2, ... , Z = 26 unless the question provides another convention. A pair can encode a fixed gap: AD, BE, and CF each move forward three positions within the pair. DH moves four positions and is odd under that rule.
Opposite letters have positions that sum to 27: A–Z, B–Y, C–X, D–W, E–V, and so on. The pair CY is not an opposite pair because C = 3 and Y = 25, giving 28. For a three-letter group, check all internal gaps: ACE, BDF, and CEG each move $+2,+2$.
Letter questions may instead use vowel membership, prime-numbered positions, reversed positions, alphabetical order, or case-sensitive shape. Do not assume that a printed number beside a letter is its forward position: it might be a reverse position, calculated as $27-p$ where $p$ is the forward position.
6. Word and general-knowledge classification
Common word categories include animals, plants, occupations, tools, materials, days, months, and places. The question should support a specific relationship among three items. In Monday, Tuesday, Friday, March, the first three are weekdays and March is a month.
Do not silently switch criteria. “Three start with a consonant” is not a convincing answer if a more direct category is clearly present. For a factual category, check the exact claim: granite is a rock, while copper, iron, and gold are metals. Avoid relying on a disputed everyday label when a better defined example is available.
Word pairs can also be classified by meaning. In dog:kennel, bird:nest, bee:hive, the second word names a usual dwelling associated with the first. lion:stable breaks that relation. This is stronger than grouping all eight printed words by an unrelated feature such as word length.
7. Mismatched relationships and analogies
Many “heterogeneous item” questions show pairs rather than individual objects. First name the relation and its direction:
- Part → whole: leaf : tree; page : book; finger : hand.
- Worker → material: tailor : cloth; potter : clay; sculptor : stone.
- Object → action: pen : write; scissors : cut; oven : bake.
- Object → class: Moon : satellite; Mercury : planet.
House : room reverses the part-to-whole direction because a room is part of a house. A pair may use individually related words and still be wrong because the direction or the kind of relationship differs. Check both halves of every option, not just whether the terms are familiar together.
For a numeric or letter analogy, write the operation explicitly. “First number cubed gives the second” is testable for 2:8, 3:27, and 4:64. It fails for 5:100 because $5^3=125$.
8. Figure and nonverbal classification
When actual diagrams are unavailable, a question may describe each figure in words. Use the same measurable attribute for all choices:
- Boundary: triangle, rectangle, and pentagon have straight sides; a circle has a curved boundary.
- Number of sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6.
- Reflection symmetry: a square, a regular pentagon, and a non-square rectangle have at least one line of symmetry; a nondegenerate scalene triangle has none.
- Curved surface in a solid: sphere, cylinder, and cone have curved surface portions; a cube has only plane faces.
- Rotational symmetry: a square matches itself after a $90^\circ$ rotation about its centre; a non-square rectangle does not.
Pay attention to qualifiers. A regular pentagon has reflection symmetry; an arbitrary pentagon need not. A rectangle rotated through $180^\circ$ matches itself, whereas a non-square rectangle rotated through $90^\circ$ does not. In a real figure question, inspect the drawing rather than assuming every picture of a named shape is regular.
9. Mixed items and multi-step rules
Some options combine letters and numbers. In A1, C3, E5, the digit matches the letter's forward position. G8 breaks it because G = 7. Check both components; a correct-looking letter progression alone can hide the mismatch.
Another mixed option may encode an arithmetic expression, a word and its category, or a place and its type. Parse the complete item as an ordered pair. If the shared rule has two requirements, such as “both digit gaps are $+1,+2$,” test both requirements for all matching choices.
Use the simplest rule that explains three choices without inventing an exception. A complicated formula can be fitted to a short list of numbers, but it is weak exam reasoning unless the prompt or options support it.
10. Common traps and ambiguity checks
Mistaking a visible difference for the rule. Every item differs from every other in some way. An odd answer needs a property common to all remaining entries.
Confusing odd term and correction. In a mismatched pair, report the printed wrong pair when asked for the odd item. If asked to repair it, also state what should replace it.
Ignoring order. Leaf:tree is part-to-whole; tree:leaf is whole-to-part. The direction matters in classification of pairs.
Overgeneralizing a figure. A scalene triangle has no reflection line; an isosceles triangle normally has one. A non-square rectangle needs the word “non-square” to make a $90^\circ$ rotation test decisive.
Forcing a unique answer. Suppose a set is 2, 3, 4, 9 with no instruction. You could single out 2 as the only even prime, 4 as the only even composite, or 9 as the only odd composite square. The choices do not fix one rule. A well-formed exam question should add a condition or use less overlapping examples.
11. Fully worked examples
Worked example 1 — common word class
Question: Violin, flute, telescope, trumpet. Find the odd item.
Work: Violin, flute, and trumpet are musical instruments. A telescope is an optical instrument, so it does not share the more specific musical-instrument class. Answer: telescope.
Worked example 2 — days and months
Question: Monday, Wednesday, April, Friday.
Work: Monday, Wednesday, and Friday are days of the week. April is a month. Answer: April.
Worked example 3 — perfect squares
Question: 16, 25, 36, 42.
Work: $16=4^2$, $25=5^2$, and $36=6^2$. No integer squared equals 42. Answer: 42.
Worked example 4 — prime numbers
Question: 11, 13, 15, 17.
Work: 11, 13, and 17 have no positive divisors other than 1 and themselves. $15=3\times5$ is composite. Answer: 15.
Worked example 5 — divisibility
Question: 14, 21, 28, 34.
Work: $14=2\times7$, $21=3\times7$, and $28=4\times7$. 34 is not a multiple of 7. Answer: 34.
Worked example 6 — digit gaps
Question: 124, 235, 346, 358.
Work: The first three numbers have digit gaps $+1,+2$: $1\to2\to4$, $2\to3\to5$, and $3\to4\to6$. In 358, the gaps are $+2,+3$. Answer: 358.
Worked example 7 — prime letter positions
Question: B, C, E, I.
Work: B = 2, C = 3, and E = 5 are prime-numbered alphabet positions. I = 9 is composite. Answer: I.
Worked example 8 — a constant letter gap
Question: AD, BE, CF, DH.
Work: The first three pairs have a $+3$ move from first to second letter: $1\to4$, $2\to5$, $3\to6$. DH moves from 4 to 8, a $+4$ gap. Answer: DH.
Worked example 9 — opposite-letter pairs
Question: AZ, BY, CX, DY.
Work: A–Z, B–Y, and C–X each have letter positions summing to 27. D = 4 and Y = 25 sum to 29; D's opposite is W. Answer: DY.
Worked example 10 — a mismatched worker and material
Question: Tailor:cloth, potter:clay, sculptor:stone, carpenter:flour.
Work: The first three pair a worker with a usual material they work. A carpenter works with wood, not flour. Answer: carpenter:flour.
Worked example 11 — relationship direction
Question: Leaf:tree, page:book, finger:hand, house:room.
Work: The first three run from part to whole. A room is part of a house, so house:room runs whole to part. Answer: house:room.
Worked example 12 — figure sides
Question: Triangle, quadrilateral, pentagon, circle. Select the one without straight sides.
Work: The first three are polygons with straight sides. A circle has a curved boundary and no straight side. Answer: circle.
Worked example 13 — a solid's surface
Question: Sphere, cylinder, cone, cube. Select the solid without a curved surface.
Work: Sphere, cylinder, and cone each have a curved surface portion. A cube's surface consists only of plane square faces. Answer: cube.
Worked example 14 — a paired number rule
Question: 3:12, 4:20, 5:30, 6:42, 7:49. Find the odd pair.
Work: The first four follow $n\to n(n+1)$: $3\times4=12$, $4\times5=20$, $5\times6=30$, and $6\times7=42$. For 7, the second number should be $7\times8=56$. Answer: 7:49.
Worked example 15 — an ambiguous list
Question: 2, 3, 4, 9. Can one odd number be proved from the list alone?
Work: No. 2 is the only even prime; 4 is the only even composite; 9 is the only odd composite square. Each description selects a different number. A condition such as “select the only odd perfect square” would make 9 the answer. Answer: no unique odd item without an added condition.
12. Practice: 30 MCQs
In each question, choose the one mismatched option. For letters, use A = 1 through Z = 26. For figure rotations, rotate around the figure's centre in its plane.
Practice 01
Which number is not a perfect square?
A. 49
B. 64
C. 90
D. 81
Practice 02
Which number is not a perfect cube?
A. 100
B. 8
C. 27
D. 64
Practice 03
Which number is composite while the others are prime?
A. 29
B. 31
C. 37
D. 39
Practice 04
Which number is not a multiple of 7?
A. 14
B. 21
C. 36
D. 28
Practice 05
In three pairs the second number equals the first number multiplied by its successor. Which pair does not fit?
A. 4:20
B. 10:100
C. 6:42
D. 8:72
Practice 06
In three numbers, the second digit is one greater than the first and the third digit is two greater than the second. Which is odd?
A. 124
B. 235
C. 348
D. 457
Practice 07
In three letter pairs, the second letter is three alphabet positions after the first. Which pair is odd?
A. AD
B. BE
C. DH
D. CF
Practice 08
Which pair does not consist of opposite letters whose positions total 27?
A. AZ
B. CY
C. DW
D. EV
Practice 09
In three groups, each successive letter moves forward two positions. Which group is odd?
A. ACE
B. BDF
C. CEG
D. DFJ
Practice 10
Which letter does not occupy a prime-numbered forward position?
A. I
B. B
C. C
D. E
Practice 11
In three letter:number pairs, the number is the letter's reverse alphabet position, with Z = 1. Which pair is odd?
A. A:26
B. C:24
C. J:17
D. M:13
Practice 12
The vowel order is A, E, I, O, U. Which pair is not made of consecutive vowels in that order?
A. AE
B. EI
C. IO
D. AU
Practice 13
Which item is outside the group of musical instruments?
A. Flute
B. Violin
C. Telescope
D. Trumpet
Practice 14
Which animal is not a feline?
A. Leopard
B. Eagle
C. Lion
D. Tiger
Practice 15
Which item is not a metal?
A. Granite
B. Copper
C. Iron
D. Gold
Practice 16
Which object:class pair is mismatched?
A. Pacific : continent
B. Mercury : planet
C. Sirius : star
D. Moon : natural satellite
Practice 17
Which animal:dwelling pair is mismatched?
A. Dog : kennel
B. Lion : stable
C. Bird : nest
D. Bee : hive
Practice 18
Which object:usual action pair is mismatched?
A. Pen : write
B. Scissors : cut
C. Hammer : sew
D. Oven : bake
Practice 19
Three pairs run from part to whole. Which pair reverses that direction?
A. House : room
B. Leaf : tree
C. Page : book
D. Finger : hand
Practice 20
Which polygon:number of sides pair is incorrect?
A. Triangle : 3
B. Pentagon : 6
C. Quadrilateral : 4
D. Hexagon : 6
Practice 21
Which figure has no line of reflection symmetry?
A. Square
B. Regular pentagon
C. Nondegenerate scalene triangle
D. Non-square rectangle
Practice 22
Which figure has a curved boundary rather than only straight sides?
A. Triangle
B. Rectangle
C. Pentagon
D. Circle
Practice 23
Which solid has no curved surface portion?
A. Sphere
B. Cube
C. Cylinder
D. Cone
Practice 24
Which stated rotation does not leave the figure looking identical?
A. Square rotated $90^\circ$
B. Equilateral triangle rotated $120^\circ$
C. Non-square rectangle rotated $180^\circ$
D. Non-square rectangle rotated $90^\circ$
Practice 25
In three pairs, the second number is the cube of the first. Which pair is odd?
A. 2:8
B. 3:27
C. 5:100
D. 4:64
Practice 26
In three pairs, the number after the colon equals the sum of the two digits before it. Which pair is odd?
A. 13:4
B. 46:12
C. 24:6
D. 35:8
Practice 27
Which worker:usual material pair is mismatched?
A. Carpenter : flour
B. Tailor : cloth
C. Potter : clay
D. Sculptor : stone
Practice 28
In three items, the printed number equals the letter's forward position. Which item is odd?
A. G8
B. A1
C. C3
D. E5
Practice 29
Three expressions equal 20. Which one has a different value?
A. $25%$ of 80
B. One quarter of 80
C. One half of 80
D. 20
Practice 30
In three triples, each next number is twice the previous one. Which triple is odd?
A. (2, 4, 8)
B. (7, 14, 21)
C. (3, 6, 12)
D. (5, 10, 20)
13. Answers and explanations
Answer 01
C. 90. The other choices are $49=7^2$, $64=8^2$, and $81=9^2$. No integer squared is 90, so it is the sole non-square.
Answer 02
A. 100. The others are $8=2^3$, $27=3^3$, and $64=4^3$. 100 lies between $4^3=64$ and $5^3=125$ and is not a perfect cube.
Answer 03
D. 39. $39=3\times13$, so it is composite. The other three numbers, 29, 31, and 37, are prime.
Answer 04
C. 36. $14=2\times7$, $21=3\times7$, and $28=4\times7$. 36 is not divisible by 7.
Answer 05
B. 10:100. The rule is $n\to n(n+1)$: $4\times5=20$, $6\times7=42$, and $8\times9=72$. For 10 the required second number is $10\times11=110$, not 100.
Answer 06
C. 348. In 124, 235, and 457, the digit gaps are $+1$ then $+2$. In 348, $3\to4$ is $+1$ but $4\to8$ is $+4$.
Answer 07
C. DH. AD is $1\to4$, BE is $2\to5$, and CF is $3\to6$, each a $+3$ shift. DH is $4\to8$, a $+4$ shift.
Answer 08
B. CY. The valid opposite pairs have position sums 27: A+Z = $1+26$, D+W = $4+23$, E+V = $5+22$. C+Y = $3+25=28$; C's opposite is X.
Answer 09
D. DFJ. ACE has positions 1,3,5; BDF has 2,4,6; CEG has 3,5,7. Each uses two $+2$ steps. DFJ has 4,6,10, so its last gap is $+4$.
Answer 10
A. I. B = 2, C = 3, and E = 5 are prime-numbered positions. I = 9 is composite because $9=3\times3$.
Answer 11
D. M:13. Reverse position is $27-p$. A has 26, C has 24, and J has 17. M is forward position 13, so its reverse position is 14, not 13.
Answer 12
D. AU. AE, EI, and IO are adjacent pairs in the stated vowel order A, E, I, O, U. A and U are the first and last vowels, not consecutive.
Answer 13
C. Telescope. Flute, violin, and trumpet are musical instruments. A telescope is an optical instrument rather than a musical one.
Answer 14
B. Eagle. Leopard, lion, and tiger are felines. Eagle is a bird and does not belong to that animal family.
Answer 15
A. Granite. Copper, iron, and gold are metals. Granite is a rock, so it does not share their class.
Answer 16
A. Pacific : continent. Mercury is a planet, Sirius is a star, and the Moon is Earth's natural satellite. The Pacific is an ocean, not a continent.
Answer 17
B. Lion : stable. A kennel is associated with dogs, a nest with birds, and a hive with bees. A stable is associated with horses, whereas a lion's shelter is commonly called a den.
Answer 18
C. Hammer : sew. A pen is used to write, scissors to cut, and an oven to bake. A hammer is used for actions such as striking or driving nails, not sewing.
Answer 19
A. House : room. A leaf is part of a tree, a page part of a book, and a finger part of a hand. A room is part of a house, so house:room runs whole to part.
Answer 20
B. Pentagon : 6. A triangle has 3 sides, a quadrilateral 4, and a hexagon 6. A pentagon has 5 sides, not 6.
Answer 21
C. Nondegenerate scalene triangle. The square, regular pentagon, and non-square rectangle each have at least one line of reflection symmetry. A scalene triangle has three unequal sides and no reflection line.
Answer 22
D. Circle. Triangle, rectangle, and pentagon are polygons with straight sides. A circle has a curved boundary.
Answer 23
B. Cube. Sphere, cylinder, and cone each have a curved surface portion. A cube's six faces are plane squares, so it has no curved surface.
Answer 24
D. Non-square rectangle rotated $90^\circ$. A square has quarter-turn symmetry, an equilateral triangle has $120^\circ$ rotational symmetry, and any rectangle has half-turn symmetry. A non-square rectangle's width and height exchange after a quarter-turn, so it does not match its original orientation.
Answer 25
C. 5:100. $2^3=8$, $3^3=27$, and $4^3=64$. The cube of 5 is 125, not 100.
Answer 26
B. 46:12. The other sums are $1+3=4$, $2+4=6$, and $3+5=8$. For 46, $4+6=10$, not 12.
Answer 27
A. Carpenter : flour. A tailor works with cloth, a potter with clay, and a sculptor may work with stone. A carpenter's usual material is wood; flour does not fit the worker-to-material relationship.
Answer 28
A. G8. A = 1, C = 3, and E = 5 match their printed numbers. G is alphabet position 7, not 8.
Answer 29
C. One half of 80. $25%$ of 80 is 20, one quarter of 80 is 20, and the standalone number is 20. One half of 80 is 40.
Answer 30
B. (7, 14, 21). The matching triples double twice: $2\to4\to8$, $3\to6\to12$, and $5\to10\to20$. In the odd triple, $7\to14$ doubles but $14\to21$ does not; doubling would give 28.
14. Quick revision
Classification requires a single stated property shared by the matching entries. For numbers, check factors, squares, cubes, digit relations, and operations in ordered pairs or triples. For letters, use alphabet positions and verify every gap. For words, identify the exact category or relationship. For figures, count sides and examine curvature, reflection, and rotation. In pair questions, preserve the direction of the relation.
Before choosing, verify that the other three options satisfy your rule and that the chosen one fails it. If several equally clear rules produce different odd answers, the data need a further condition.
This is AI-generated information.