Logical / Abstract Reasoning

Reasoning · Analytical puzzles · Topic 1 of 14

Number Series

Mark as done

Think of it like…

Your gullak (piggy bank) savings

Suppose you drop coins into a clay gullak every evening and write the total in a notebook: ₹3, ₹8, ₹15, ₹24, ₹35. Looking only at the totals, the pattern is not obvious. But ask “how much did I add each day?” and you see ₹5, ₹7, ₹9, ₹11 — every day you add ₹2 more than the day before. So tomorrow you will add ₹13 and the jar will hold ₹48. On another month, Aai doubles whatever is in your jar every Sunday; then the totals grow by multiplying, not adding. And if you and your sister fill two jars on alternate days, the notebook mixes two separate lists. A number series question is exactly this notebook: find what was added (or multiplied) each step, and the next total writes itself.

In everyday life	In the exam topic
Total in the jar each evening	Term of the series
Money added each day	First difference (next term − this term)
“I add ₹2 more than yesterday”	Constant second difference
Aai doubling the savings every Sunday	Multiplication / ratio pattern (×2, ×2 + 1 …)
You and your sister writing on alternate days	Alternating series: two series mixed together

The concept

A number series is a list of numbers made by a hidden rule. Your job is to find that rule and use it for the next term (or to catch the one term that breaks it). Always start by writing the differences between neighbouring terms below the series. If the differences are equal, it is an arithmetic series. If the differences themselves change by a fixed amount, look at the second differences.

If the differences grow very fast (roughly doubling or tripling), test ratios instead: ×2, ×3, or mixed rules like ×2 + 1 or ×3 − 2. If the terms are close to familiar numbers, compare them with squares (1, 4, 9, 16, 25 …) and cubes (1, 8, 27, 64, 125 …): many series are n² ± 1, n³ + n, or n(n + 1).

If nothing fits, split the series into odd and even positions. Often two simple series are woven together (alternating series). For “find the wrong number” questions, find the rule from the majority of terms, then check each term against it one by one.

Key points & formulas

Step order: differences → second differences → ratios → squares/cubes → alternate positions
Equal differences: arithmetic series (a, a + d, a + 2d …)
Differences 2, 4, 6, 8 … or 5, 7, 9, 11 …: second difference is constant
Fast growth: test ×k, ×k + c, ×k − c (e.g. 5, 11, 23, 47 is ×2 + 1)
Know squares to 30² and cubes to 15³ by heart
n(n + 1): 2, 6, 12, 20, 30 …; n² − 1: 3, 8, 15, 24 …; n³ + n: 2, 10, 30, 68 …
Long series (7–8 terms) with a zig-zag: check odd and even positions separately
Wrong-number questions: the wrong term usually makes two consecutive differences look odd

Worked example

Find the next term: 3, 8, 15, 24, 35, ?

Show solution

First differences: 8 − 3 = 5, 15 − 8 = 7, 24 − 15 = 9, 35 − 24 = 11.
Differences 5, 7, 9, 11 rise by 2 each time, so the next difference is 13.
Next term = 35 + 13 = 48.
Check with squares: the terms are 2² − 1, 3² − 1, 4² − 1, 5² − 1, 6² − 1, so the next is 7² − 1 = 48. Both methods agree.

Answer: 48

Exam shortcut

Look at the last digits and the size jump first. If a term roughly doubles each time, go straight to ×2 ± something and skip the difference table. If the terms sit just above or below squares or cubes, test n² ± 1 or n³ ± n before anything else.

Common traps

Stopping after checking only two terms. A rule must fit every gap in the series, not just the first one.
Missing an alternating series: if the differences jump up and down (e.g. +3, +1, +6, +2), split the series into two.

Practice · tap an option to check

Q1. Find the next term: 5, 11, 23, 47, 95, ?

Q2. Find the next term: 2, 10, 30, 68, 130, ?

Q3. Find the next term: 4, 7, 8, 14, 16, 21, 32, ?

Q4. Find the wrong number in the series: 3, 5, 9, 17, 33, 66, 129

↑ Back to topic list

Reasoning · Analytical puzzles · Topic 2 of 14

Letter & Alphanumeric Series

Mark as done

Think of it like…

House numbers on a circular society lane

Picture a housing society where the lane has exactly 26 bungalows, numbered 1 to 26, and the lane goes round in a loop so that after bungalow 26 you are back at bungalow 1. Name the bungalows A to Z. Now “move 3 letters ahead of W” simply means “walk 3 houses ahead of house 23”: 24, 25, 26 — that is Z. Walk one more and you reach house 1, A, because the lane loops. Bungalow 1 (A) faces bungalow 26 (Z) across the garden, B faces Y, and so on: every pair of facing houses adds up to 27. Once you think in house numbers, a letter series becomes a number series you already know how to solve.

In everyday life	In the exam topic
Bungalow number on the lane	Position of the letter (A = 1 … Z = 26)
Walking k houses ahead	Moving k letters forward (+k)
Lane loops back to house 1 after 26	After Z comes A again (subtract 26)
House facing yours across the garden	Opposite letter: positions add to 27 (A–Z, B–Y …)

The concept

Convert every letter into its position number (A = 1, B = 2 … Z = 26), solve the number pattern, then convert back. To get positions quickly, remember EJOTY = 5, 10, 15, 20, 25. Every other letter is one or two steps from these anchors: R is 2 before T, so R = 18.

For backward positions use 27 − position: the 10th letter from the right (Z end) is 27 − 10 = 17 = Q. The opposite letter of any letter is also 27 − position (G = 7 ↔ 27 − 7 = 20 = T). If a count goes past 26, subtract 26; if it goes below 1, add 26.

In letter-cluster series (AZ, CX, EV …), treat each place in the cluster as its own series. In alphanumeric arrangements (a mixed string of letters, digits and symbols), read the question very carefully: “immediately preceded by” means the element just before it, “immediately followed by” means the one just after it. Count slowly and mark each match on paper.

Key points & formulas

EJOTY: E = 5, J = 10, O = 15, T = 20, Y = 25
Position from the right end = 27 − position from the left
Opposite pairs add to 27: A–Z, B–Y, C–X, D–W, E–V, F–U, G–T, H–S, I–R, J–Q, K–P, L–O, M–N
Moving forward beyond Z: subtract 26; moving back beyond A: add 26
“k-th to the left of X” = position of X − k; “k-th to the right” = position + k
Clusters: solve the 1st letters, 2nd letters (and digits) as separate series
Vowels A, E, I, O, U are at 1, 5, 9, 15, 21

Worked example

Find the next letter: B, E, I, N, ?

Show solution

Convert to positions: B = 2, E = 5, I = 9, N = 14.
Differences: +3, +4, +5. The next difference is +6.
14 + 6 = 20.
Position 20 = T (from EJOTY).

Answer: T

Exam shortcut

Never count on your fingers from A. Jump to the nearest EJOTY anchor and count one or two steps: to find the 22nd letter, start at T (20) and step U (21), V (22).

Common traps

Confusing “from the left” and “from the right”. The 10th letter from the right is Q (27 − 10), not J.
In alphanumeric strings, symbols like #, @, $ are neither letters nor numbers — do not count them as either.

Practice · tap an option to check

Q1. Find the next term: AZ, CX, EV, GT, ?

Q2. Find the next letter: Z, W, S, N, ?

Q3. In the arrangement M 4 # K 7 R @ 2 E 9 $ B 5 T, how many letters are immediately preceded by a number?

Q4. In the English alphabet, which letter is 6th to the left of the 10th letter from the right end?

↑ Back to topic list

Reasoning · Logical deduction · Topic 3 of 14

Coding–Decoding

Mark as done

Think of it like…

The Mumbai dabbawala tiffin code

Mumbai’s dabbawalas deliver lakhs of lunch boxes using a few painted letters, numbers and colours on each tiffin lid. A new dabbawala does not memorise every box. He learns the rule: this mark means the starting station, that one means the building, this number means the floor. Once he cracks the rule from a few known tiffins, he can read any new tiffin. And if two tiffins both go to Churchgate, the mark they share must be the Churchgate mark. Coding–decoding questions work the same way: they show you one or two coded examples, you discover the rule by comparing, and then you apply that same rule to a new word.

In everyday life	In the exam topic
A painted code on a tiffin lid	The coded word or sentence
The one rule all dabbawalas follow	The coding rule (shift, reverse, opposite letter, number value)
Two tiffins going to the same station share a mark	A word common to two sentences gets the code common to both
A new dabbawala reading a new tiffin	Applying the found rule to the word asked in the question

The concept

Letter coding: write the position numbers of the original word above the code and find the change letter by letter. Common rules: every letter shifted by a fixed amount (+1, −2), shifts that grow (+1, +2, +3 …), the word reversed, or each letter replaced by its opposite letter (positions adding to 27).

Number coding: the code is often the sum of letter positions, the number of letters × something, or each letter replaced by its position. Test the simplest idea (sum of positions) first.

Sentence (word) coding: compare two sentences at a time. The word common to both sentences must have the code common to both coded forms. Eliminate step by step until only the asked word is left. Substitution coding (“sky is called sea …”) is a trap of language: first answer the real-world question, then replace that real answer with its code name.

Key points & formulas

Always write positions above and below: compare letter by letter
Check for a constant shift first, then reversal, then opposite letters (sum 27)
Opposite-letter coding: GIRL → TRIO because G↔T, I↔R, R↔I, L↔O
Number codes: try sum of positions, then product, then number of letters
Sentence coding: common word ↔ common code; use pairs of sentences
Substitution: answer the real question first, then give its code name
If the code is shorter or longer than the word, letters are being dropped, added or grouped

Worked example

In a code language, “pit na sa” means “you are good”, “na ka da” means “they are bad”, and “sa ri ka” means “good and bad”. What is the code for “they”?

Show solution

Sentences 1 and 2 share the word “are” and the code “na”. So are = na.
Sentences 1 and 3 share “good” and the code “sa”. So good = sa.
Sentences 2 and 3 share “bad” and the code “ka”. So bad = ka.
Sentence 2 is “na ka da” = “they are bad”. With are = na and bad = ka, the only code left is “da”.

Answer: they = da

Exam shortcut

For shift codes, check only the first and last letters. If both shift by the same amount, the rest almost always follow — then pick the option whose first and last letters match, and verify one middle letter.

Common traps

In substitution questions, students write the real word (“water”) instead of its code name. Read the chain again and give the name it is “called”.
Assuming the order of codes matches the order of words. In sentence coding the codes can be jumbled; only common words and common codes matter.

Practice · tap an option to check

Q1. If PAPER is coded as QBQFS, how is PENCIL coded in the same language?

Q2. If GIRL is coded as TRIO, how is BOY coded?

Q3. If CAT = 24, then DOG = ?

Q4. If “sky” is called “sea”, “sea” is called “water”, “water” is called “air”, and “air” is called “cloud”, then where do fish live?

↑ Back to topic list

Reasoning · Logical deduction · Topic 4 of 14

Blood Relations

Mark as done

Think of it like…

The family photo at a wedding

At a Maharashtrian wedding, the photographer calls “sagle family, ekatra ya!” and arranges the whole family for one big photo. Aaji–Ajoba sit in the front centre, their sons and daughters with their spouses stand in the next row, and the grandchildren squat in front. Couples always stand side by side; children stand below their parents. With this photo in hand you can answer any “How is she related to him?” question at a glance — you do not need to remember who said what at the haldi. A blood relation question is a list of introductions; your job is to draw that photo on paper before answering.

In everyday life	In the exam topic
Rows in the wedding photo (Aaji–Ajoba, parents, kids)	Generations: draw each on its own level
A couple standing side by side	Horizontal double line (=) for husband–wife
Children standing below their parents	Vertical line from parent down to child
Gents in kurta, ladies in saree	+ for male, − for female next to each name
Looking at the finished photo to answer	Reading the relation off the tree, not from memory

The concept

Draw a family tree for every question. Mark each person with + (male) or − (female) as soon as the gender is known. Put people of the same generation on the same level. Join husband and wife with a horizontal double line (=), brothers and sisters with a single horizontal line, and draw a vertical line from parent down to child.

For “pointing to a photo” statements, break the sentence from the end. In “She is the daughter of the only son of my father’s father”, first decode “the only son of my father’s father” (= my father), then “daughter of my father” (= my sister). Replace each phrase with one person before moving on.

In coded relations (P + Q means P is the father of Q, etc.), translate each symbol into a line on the tree, one symbol at a time from left to right. Never assume a person’s gender unless a word (brother, wife, son) or a symbol tells you.

Key points & formulas

Symbols: + male, − female, = married couple, | parent to child, — siblings
Same generation on the same horizontal level
Father’s or mother’s brother = uncle; their sister = aunt; their children = cousins
Son’s wife = daughter-in-law; daughter’s husband = son-in-law
Wife’s or husband’s brother = brother-in-law; sister’s husband = brother-in-law
“Only son of my father’s father” = my father; “only daughter of my mother” = me (if I am female)
Gender is unknown unless stated — “A is the child of B” says nothing about A’s gender
Decode long “pointing” sentences from the end backwards

Worked example

Pointing to a photograph, Ravi says, “She is the daughter of the only son of my father’s father.” How is the girl in the photograph related to Ravi?

Show solution

Start from the end: “my father’s father” = Ravi’s paternal grandfather.
“The only son of my father’s father” = Ravi’s father (the grandfather has only one son, and Ravi’s father is his son, so it must be him).
“The daughter of my father” = a daughter of Ravi’s father.
Ravi is male, so a daughter of Ravi’s father is not Ravi himself — she is Ravi’s sister.

Answer: She is Ravi’s sister.

Exam shortcut

In “pointing” questions, cross out each phrase and write the single relation above it: “only son of my father’s father” → father, “daughter of father” → sister. Three phrases usually shrink to one word in ten seconds.

Common traps

Assuming gender from a name or a word like “child”. “B is the sister of C” tells you B is female, but not C.
“A is the brother of B” and “B is the brother of A” are not the same fact — the second also tells you B is male.

Practice · tap an option to check

Q1. A is the brother of B. B is the sister of C. C is the father of D. How is A related to D?

Q2. Pointing to a man, Neha says, “His mother is the only daughter of my mother.” How is Neha related to the man?

Q3. P + Q means P is the father of Q; P − Q means P is the wife of Q; P × Q means P is the brother of Q. If A + B × C − D, how is D related to A?

Q4. M is the son of N. N is the daughter of O. O is the husband of P. How is P related to M?

↑ Back to topic list

Reasoning · Analytical puzzles · Topic 5 of 14

Direction Sense

Mark as done

Think of it like…

Directions from the ST stand to your house in the village

A friend from the city gets off the ST bus in your village and phones you. You say: “Walk straight towards the Hanuman temple, about 4 km. At the temple turn right and go 3 km to the well. Our house is right there.” If your friend sketches this on paper — the bus stand as a dot, an arrow towards the temple, a right turn — he can tell how far your house is from the stand in a straight line, even though he walked 7 km on the roads. He also knows that in the morning the sun rises behind the temple (east), so shadows fall the other way. Direction questions are exactly this sketch: a start point, straight roads, turns, and a final straight-line distance.

In everyday life	In the exam topic
The ST bus stand where your friend gets down	Starting point at the origin (0, 0)
“Turn right at the temple”	Turn 90° clockwise from the current facing direction
Sun rising behind the temple in the morning	East reference: morning shadows fall to the west
Straight-line distance from the stand to your house	Displacement = √(x² + y²) by Pythagoras

The concept

Draw a cross for North (up), South (down), East (right) and West (left) at the start. Then trace the path on paper, writing the length on each segment. Turns depend on which way the person is currently facing: a right turn is 90° clockwise and a left turn is 90° anticlockwise from the walker’s direction, not from the page.

For the final distance, add up the net east–west movement (x) and the net north–south movement (y), then use Pythagoras: distance = √(x² + y²). The answer direction is read from the start point to the end point (e.g. x positive and y negative means South-East). Common triples: 3-4-5, 6-8-10, 5-12-13, 8-15-17.

For shadow questions: in the morning the sun is in the east, so shadows fall towards the west; in the evening the sun is in the west, so shadows fall towards the east. At noon there is almost no shadow (do not use noon in questions). If the shadow falls to your left in the morning, your left is west, so you are facing north.

Key points & formulas

Right turn = 90° clockwise; left turn = 90° anticlockwise (from current facing)
Clockwise order: N → E → S → W → N; 45° turns: N → NE → E → SE → S …
Net x = East − West; net y = North − South; distance = √(x² + y²)
Triples: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 7-24-25
Morning: sun in east, shadow to the west. Evening: sun in west, shadow to the east
“Direction of A from B” means stand at B and look towards A
A U-turn is 180°; two right turns or two left turns also reverse the direction

Worked example

Ramesh walks 10 m north, turns right and walks 6 m, then turns right again and walks 18 m. How far and in which direction is he from his starting point?

Show solution

Start at (0, 0) facing north. Walk 10 m → (0, 10).
Facing north, a right turn makes him face east. Walk 6 m → (6, 10).
Facing east, a right turn makes him face south. Walk 18 m → (6, 10 − 18) = (6, −8).
Distance = √(6² + 8²) = √100 = 10 m.
He is to the east (x = +6) and south (y = −8) of the start: South-East.

Answer: 10 m, South-East of the starting point

Exam shortcut

Ignore the order of moves when finding the final position: just total all north moves minus south moves, and all east moves minus west moves. You only need the turns to know which direction each move was in.

Common traps

Turning according to the page instead of the walker. If he is facing south, his “right” is west, not east.
Answering the direction of the start from the end when the question asks the end from the start (or the other way round). These are opposite directions.

Practice · tap an option to check

Q1. A man walks 5 km south, turns left and walks 3 km, then turns left again and walks 9 km. How far is he from his starting point?

Q2. One morning after sunrise, Suresh was standing facing a pole. The shadow of the pole fell exactly to his right. Which direction was he facing?

Q3. Priya is facing east. She turns 90° clockwise, then 180° anticlockwise, then 45° clockwise. Which direction is she facing now?

Q4. A temple is 6 km north of the bus stand. A well is 8 km east of the temple. How far is the bus stand from the well, and in which direction?

↑ Back to topic list

Reasoning · Analytical puzzles · Topic 6 of 14

Ranking & Ordering

Mark as done

Think of it like…

The queue at the ration shop

On the first of the month there is a long queue outside the ration shop (sasta dhanya dukan). You count: you are 7th from the shop window. Your friend at the back shouts that you are 12th from his end. How many people are in the queue? Counting from the front you counted yourself, and counting from the back you counted yourself again, so the total is 7 + 12 − 1 = 18. If two people swap places in the queue, the place of one tells you the place of the other. And when an aunty says “I am behind Mrs Patil but ahead of Mrs Shinde”, you can line them up without seeing the queue. Ranking questions are this queue, written in words.

In everyday life	In the exam topic
Your place from the shop window	Rank from the top / left
Your place from the back of the queue	Rank from the bottom / right
You get counted twice	Total = left + right − 1
Two people swapping places	Interchange: one person’s new position = the other’s old position
“Behind Mrs Patil, ahead of Mrs Shinde”	Ordering clues: write a chain with > signs

The concept

For one person: Total = position from left + position from right − 1. The same formula gives position from one end: position from right = Total − position from left + 1. Ranks in a class work the same way (rank from top and rank from bottom).

For two people, first convert both positions to the same end. Then the number of people between them = (difference of positions) − 1. In interchange questions, after swapping, A stands where B stood — so A’s new position from the left is B’s position from the left. Combine that with B’s position from the right to get the total.

For ordering by height, marks, age, etc., write each clue as a chain using > (taller, more, older) and merge the chains: P > Q, R > P gives R > P > Q. Place the most definite clues first. If the chain cannot be completed, the answer is “cannot be determined” — do not guess.

Key points & formulas

Total = L + R − 1
Position from right = Total − L + 1
Between two people (same end positions a and b): |a − b| − 1
Interchange: A’s new position = B’s old position (from the same end)
Ordering: build one chain with > and read the k-th from the top
Minimum total in a row when A is p-th from left and B is q-th from right (B to the right of A, no overlap) = p + q

Worked example

In a row of students, Amit is 12th from the left end and 18th from the right end. How many students are there in the row?

Show solution

Amit is counted once from the left (12) and once from the right (18).
He has been counted twice, so subtract 1.
Total = 12 + 18 − 1 = 29.

Answer: 29 students

Exam shortcut

For interchange questions, ignore the person who moved and look only at the seat: “A becomes 15th from left after taking B’s place” means B’s seat is 15th from left. Then use L + R − 1 with B’s right position.

Common traps

Forgetting the −1 in L + R − 1, giving 30 instead of 29.
Mixing ends when counting people between two persons: always convert both to positions from the same end first.

Practice · tap an option to check

Q1. In a class of 40 students, Rahul ranks 13th from the top. What is his rank from the bottom?

Q2. In a row, A is 10th from the left and B is 9th from the right. When they interchange their positions, A becomes 15th from the left. How many people are in the row?

Q3. Among P, Q, R, S and T: P is taller than Q but shorter than R. S is taller than R. T is shorter than Q. Who is the third tallest?

Q4. In a row of 30 children, Meena is 8th from the left and Kiran is 12th from the right. How many children are there between Meena and Kiran?

↑ Back to topic list

Reasoning · Analytical puzzles · Topic 7 of 14

Seating Arrangement — Linear & Circular

Mark as done

Think of it like…

Seats in a Mumbai local train compartment

In a local train, the long benches face each other: one row faces the engine and the row opposite faces backwards. If you face the engine, your right hand points to one window; the passenger sitting opposite you has his right hand pointing to the other window. So “X is to the right of Y” means a different seat depending on which bench Y sits on. Regular commuters also fix seats by one known person: “Kaka always sits by the window — I sit two seats from him.” A round table at a hotel is the same bench bent into a ring with its two ends joined. Seating questions are solved by fixing one passenger, drawing the bench, and turning “right/left” into seats according to the direction each person faces.

In everyday life	In the exam topic
Bench facing the engine vs facing backwards	People facing north vs south: left and right are swapped
“Kaka always takes the window seat”	Start with the most definite clue (fixed position)
“I sit two seats from Kaka”	Relative clues: second to the right/left
Bench bent into a ring at a round table	Circular arrangement: no ends, only neighbours and opposites
Facing the centre of the round table	Right = anticlockwise, left = clockwise

The concept

Linear: draw blanks for the seats and write which way the row faces. If people face north (up the page, the same way you are looking), their right is your right. If they face south, their right is your left. Place the fixed clues first (“C is third from the left end”, “A sits at an end”), then the relative clues (“B is to the immediate right of C”). If a clue gives two possibilities, draw both cases and kill the one that breaks a later clue.

Circular: draw a circle with the right number of seats. Fix any one person at the bottom — in a circle every seat is the same until someone sits. When people face the centre, “right” is anticlockwise and “left” is clockwise. When they face outward, it is the reverse. With n seats (n even), the person opposite is n/2 seats away.

Read every question in the set before you start: sometimes one clue is only useful later. After filling the arrangement, re-check every clue once against your final drawing — this catches most mistakes.

Key points & formulas

Facing north: their right = page right. Facing south: their right = page left
Circle facing centre: right = anticlockwise, left = clockwise
Circle facing outward: right = clockwise, left = anticlockwise
“Immediate right” = the very next seat; “second to the right” = skip one seat
n people in a circle: opposite person is n/2 seats away (n even)
Place definite clues first, then relative clues, then negative clues (“not adjacent”)
Two possible cases? Draw both side by side and eliminate

Worked example

Five friends V, W, X, Y and Z sit in a row facing north. X sits in the middle. Z sits at the extreme right end. W sits second to the left of X. Y is not an immediate neighbour of Z. Who sits between X and Z?

Show solution

Seats 1 to 5 from left to right; facing north, their left and right match the page.
X is in the middle: seat 3. Z is at the right end: seat 5.
W is second to the left of X: seat 3 − 2 = seat 1.
Seats 2 and 4 are left for V and Y. Y cannot be next to Z (seat 4), so Y is in seat 2 and V is in seat 4.
Order: W, Y, X, V, Z. The person between X (3) and Z (5) is V.

Answer: V

Exam shortcut

In a circle where everyone faces the centre, just remember “Right = Reverse of clock” (anticlockwise). Write a small ↺ = R next to your circle so you never mix it up during the exam.

Common traps

Using page-right for people facing south. They are like the passengers on the opposite bench: their right is your left.
Reading “B sits between A and C” as “immediately between” when the question does not say so. Unless it says “immediately”, there may be other people in between.

Practice · tap an option to check

Q1. Six friends A, B, C, D, E and F sit in a row facing north. C is third from the left end. B sits to the immediate right of C. A sits at one of the ends. E is an immediate neighbour of both A and B. F does not sit at an end. Who sits at the left end?

Q2. Six friends A, B, C, D, E and F sit in a row facing north. C is third from the left end. B sits to the immediate right of C. A sits at one of the ends. E is an immediate neighbour of both A and B. F does not sit at an end. Who sits second to the right of C?

Q3. Six people P, Q, R, S, T and U sit around a circular table facing the centre. S sits opposite P. Q sits to the immediate right of P. R sits second to the left of Q. U sits between Q and S, next to both. Who sits opposite R?

Q4. Six people P, Q, R, S, T and U sit around a circular table facing the centre. S sits opposite P. Q sits to the immediate right of P. R sits second to the left of Q. U sits between Q and S, next to both. Who sits second to the right of S?

↑ Back to topic list

Reasoning · Analytical puzzles · Topic 8 of 14

Puzzles — Scheduling, Floors & Multi-attribute Grids

Mark as done

Think of it like…

The Ganpati mandal volunteer roster

Every year the Ganpati mandal secretary must make a roster: who does the aarti, who handles the prasad, who manages the crowd, and on which day. Everyone has a condition — “Sachin can’t come before Wednesday”, “Pooja does prasad only”, “Rohit must come the day after Sachin”. The secretary does not keep it in his head. He draws a table with names down the side and days across the top, fills in the definite facts first (Pooja = prasad), puts ✗ where something is impossible, and soon each row and column has only one empty box left. When a condition gives two choices, he pencils in both and checks which one breaks a rule. Puzzle questions in the exam are this roster.

In everyday life	In the exam topic
Names down the side, days/duties across the top	Grid with one row per person, one column per attribute value
“Pooja does prasad only”	Definite clue: tick it first
“Sachin can’t come before Wednesday”	Negative clue: put ✗ in those boxes
One person per day, one day per person	Each row and each column gets exactly one tick
Pencilling two possible rosters	Case analysis: try both, eliminate the one that breaks a clue

The concept

Most puzzles have people matched to one or more attributes (day, floor, city, colour, subject). Draw a table: one row per position (Monday, Tuesday … or floor 1 to 5) and one column per attribute. Write every clue in short form beside the table so you do not keep re-reading the paragraph.

Fill clues in this order: (1) definite clues (“Chirag lives on the top floor”), (2) linked clues that hang on a placed person (“Amit lives two floors below Chirag”), (3) negative clues (“Joshi does not teach Python”). When a clue still allows two places, write both cases side by side and continue each until one breaks a rule.

For floor puzzles, decide the numbering first: floor 1 is the lowest unless told otherwise. “Above” and “below” do not mean “immediately”. For scheduling, “the day after” means immediately next, but “before” means any earlier day. When the grid is complete, check every clue once more.

Key points & formulas

Draw the table before reading clue 2
Order: definite → linked → negative clues
“Immediately above/after” = next one; “above/after” = any one higher/later
Each person–value pair is used exactly once (one tick per row and column)
Two cases? Solve both; the survivor is the answer
Floor 1 is the bottom unless the question says “ground floor = 0”
Re-check every clue against the final table before answering

Worked example

Three friends Om, Pooja and Rohan come from Pune, Nashik and Nagpur (one each), and each likes a different sweet: modak, puran poli and shrikhand. (1) Pooja is from Nashik. (2) The friend from Pune likes modak. (3) Rohan likes neither modak nor shrikhand. Who likes shrikhand, and which city is that person from?

Show solution

Make a 3 × 2 table: Name | City | Sweet.
Clue 1 (definite): Pooja = Nashik.
Clue 3: Rohan likes neither modak nor shrikhand, so Rohan = puran poli.
Clue 2: the Pune friend likes modak. Rohan does not like modak, so Rohan is not from Pune. Pooja is from Nashik, so Rohan = Nagpur and Om = Pune.
Om (Pune) likes modak. The only sweet left is shrikhand, for Pooja.

Answer: Pooja, from Nashik, likes shrikhand.

Exam shortcut

Read all the questions in the set first. Sometimes the question only asks about one person, and you can answer it after placing just two or three clues, without completing the whole table.

Common traps

Treating “Bina lives below Eknath” as “immediately below”. Without the word “immediately”, any lower floor is possible.
Forgetting that “before” in scheduling means any earlier day, not the previous day.

Practice · tap an option to check

Q1. Five friends Amit, Bina, Chirag, Divya and Eknath live on floors 1 to 5 of a building (floor 1 is the lowest), one per floor. Chirag lives on the top floor. Amit lives two floors below Chirag. Eknath lives immediately below Amit. Divya lives on an even-numbered floor. Who lives on floor 1?

Q2. Five friends Amit, Bina, Chirag, Divya and Eknath live on floors 1 to 5 of a building (floor 1 is the lowest), one per floor. Chirag lives on the top floor. Amit lives two floors below Chirag. Eknath lives immediately below Amit. Divya lives on an even-numbered floor. How many people live between Bina and Divya?

Q3. Four lectures — C, Java, Python and SQL — are held on Monday to Thursday, one per day, each taught by a different professor: Joshi, Kulkarni, Patil and Shaikh. Java is held the day after C. SQL is held on some day before C. Python is not on Monday or Tuesday. Patil teaches Java. Shaikh’s lecture is on Monday. Joshi does not teach Python. Who teaches the lecture on Tuesday?

Q4. Four lectures — C, Java, Python and SQL — are held on Monday to Thursday, one per day, each taught by a different professor: Joshi, Kulkarni, Patil and Shaikh. Java is held the day after C. SQL is held on some day before C. Python is not on Monday or Tuesday. Patil teaches Java. Shaikh’s lecture is on Monday. Joshi does not teach Python. Which lecture does Prof. Kulkarni teach, and on which day?

↑ Back to topic list

Reasoning · Logical deduction · Topic 9 of 14

Syllogisms

Mark as done

Think of it like…

WhatsApp groups and their members

Think of “Class FY-MCA”, “College Cricket” and “Family” as WhatsApp groups. “All FY-MCA members are in the College group” means the class group sits completely inside the college group. “Some cricket members are in FY-MCA” means at least one person is in both — maybe one, maybe all. “No family member is in College Cricket” means the two groups share nobody. Now, you cannot see the member list of a group you are not in, so you must ask: is this statement true whatever the hidden members are? If yes, it definitely follows. If it is true in some member lists and false in others, it is only a possibility. That is exactly how syllogisms are judged.

In everyday life	In the exam topic
A WhatsApp group	A category (circle in the Venn diagram)
Class group completely inside the College group	“All A are B”: circle A inside circle B
At least one common member in two groups	“Some A are B”: circles overlap (at least one element)
Two groups with no common member	“No A is B”: separate circles
Cannot see a hidden member list	Check every possible diagram: definite only if true in all

The concept

Draw the simplest Venn diagram that satisfies all the statements, and then ask: can I redraw it (still obeying every statement) so that the conclusion becomes false? If you can, the conclusion does not follow definitely. If no valid drawing makes it false, it follows. “Some” means at least one, possibly all.

Useful definite results: “All A are B” gives “Some A are B” and “Some B are A”. “Some A are B” gives “Some B are A”. “No A is B” gives “No B is A”. Two “some” statements (Some A are B, Some B are C) give no definite link between A and C. “All A are B + No B is C” gives “No A is C”. “Some A are B + No B is C” gives “Some A are not C”.

Possibility conclusions (“Some A being C is a possibility”) follow if at least one valid diagram makes them true; they fail only if the statements force them false. Either–or: when neither of a pair like “Some A are B” and “No A is B” follows definitely, but between them they cover every case (same two terms), the answer is “Either I or II follows”.

Key points & formulas

All A are B → A inside B; also Some A are B and Some B are A
Some A are B → Some B are A (at least one common)
No A is B → No B is A
All A–B + All B–C → All A are C; All A–B + No B–C → No A is C; Some A–B + All B–C → Some A are C; Some A–B + No B–C → Some A are not C
Some + Some, or No + No, or All (A in B) + Some (B with C): no definite A–C conclusion
“All B are A” does NOT follow from “All A are B” — but it is a possibility
Possibility fails only when a definite statement forbids it
Either–or: a complementary pair on the same two terms (Some/No, or All/Some-not) where neither is definite alone

Worked example

Statements: All pens are books. Some books are bags. Conclusions: I. Some pens are bags. II. Some bags are books.

Show solution

Draw the pens circle inside the books circle. Draw a bags circle overlapping the books circle.
Conclusion I: can the bags circle touch only the part of books outside pens? Yes. Then no pen is a bag, so I does not follow definitely.
Conclusion II: “Some books are bags” means at least one book is a bag, so that object is a bag that is a book. II follows (it is just the reverse of the statement).

Answer: Only II follows.

Exam shortcut

For a quick check, use the rule table: a definite conclusion linking the two end terms needs the middle term to be fully covered in at least one statement — as the subject of an “All” statement, or in a “No” statement. Here the middle term is books: “All pens are books” covers pens, not books, and “Some books are bags” does not cover books either — so no definite pen–bag conclusion follows.

Common traps

Using real-world knowledge. If the statement says “All cats are dogs”, accept it as true inside the question.
Concluding “All B are A” from “All A are B”. The class group is inside the college group; the college group is not inside the class group.

Practice · tap an option to check

Q1. Statements: All dogs are animals. All animals are living beings. Conclusions: I. All dogs are living beings. II. Some living beings are dogs.

Q2. Statements: Some teachers are singers. No singer is a dancer. Conclusions: I. Some teachers are not dancers. II. No teacher is a dancer.

Q3. Statements: Some phones are laptops. Some laptops are tablets. Conclusions: I. Some phones are tablets. II. No phone is a tablet.

Q4. Statements: All cars are buses. No bus is a truck. Conclusions: I. Some cars being trucks is a possibility. II. All buses being cars is a possibility.

↑ Back to topic list

Reasoning · Verbal reasoning · Topic 10 of 14

Statements — Assumptions, Conclusions & Inferences

Mark as done

Think of it like…

The gram panchayat’s drum announcement (dawandi)

The gram panchayat sends a man around with a drum: “Aika ho aika! During the monsoon, boil your drinking water to avoid jaundice!” Why would the sarpanch spend money on this? Because he silently believes two things: that people will listen to the announcement, and that boiling water really cuts the risk of jaundice. Those unspoken beliefs are assumptions. What can the villagers safely take away? “If I boil my water, I lower my risk” — a conclusion that comes straight from the words. But the chatter at the chavdi — “So the whole village has jaundice now!” or “Only boiled water is safe anywhere in India” — goes beyond what was announced. Exam questions are this drum announcement: you judge what is behind it and what follows from it, using only the words announced.

In everyday life	In the exam topic
The drum announcement	The statement or passage
Why the sarpanch bothered to announce it	Implicit assumption: taken for granted, not written
What a villager can safely take away	Conclusion: follows directly from the statement
Exaggerated gossip at the chavdi	Invalid: extreme words, outside knowledge, goes beyond the text

The concept

An assumption is something the speaker must believe for the statement to make sense, but has not said. Test it by negation: if the assumption were false, would the statement become pointless? (If people never listen to announcements, the drum is useless — so “people listen” is an assumption.) An advertisement assumes people want what it promises; an appeal assumes people can act on it.

A conclusion must follow from the statement alone, using no outside facts. Reject conclusions with extreme words (only, all, always, never, definitely) unless the statement itself uses them. An inference from a passage is what can reasonably be deduced from the facts given; options that add new facts, predict the future, or generalise beyond the passage are wrong.

Work option by option, and keep your own opinions out. Something can be true in real life and still not follow from the statement. Answer choices are usually: only I, only II, both, neither (sometimes either). Decide I and II separately before looking at the options.

Key points & formulas

Assumption = unstated belief the statement depends on
Negation test: if the assumption is false, does the statement lose its point?
Conclusion = must follow from the given words only
Extreme words (only, all, never, always, best) usually make a choice wrong
Advice/appeal assumes the advice is useful and people can follow it
Inference questions: no new facts, no future predictions, no generalising
Real-world truth is irrelevant — only the statement matters

Worked example

Statement: The gram panchayat announced, “Boil drinking water during the monsoon to avoid jaundice.” Assumptions: I. People pay attention to the panchayat’s announcements. II. Boiling water helps reduce the risk of jaundice.

Show solution

Assumption I: if nobody listened, the announcement would be pointless. So I is implicit.
Assumption II: if boiling made no difference, the panchayat would not ask for it. So II is implicit.
Neither assumption uses an extreme word or an outside fact.

Answer: Both I and II are implicit.

Exam shortcut

For every assumption or conclusion, quickly underline any extreme word (only, all, never, always, best, definitely). If the statement does not contain that strength, mark the option invalid at once — this settles about half the choices in seconds.

Common traps

Choosing an option because it is true in real life. The only question is whether it follows from (or is assumed by) the statement.
Treating a restatement as an assumption. An assumption is unsaid; something already written in the statement is not an assumption.

Practice · tap an option to check

Q1. Statement: “Drink Shakti tea — stay fresh and full of energy all day!” — an advertisement. Assumptions: I. People want to feel fresh and energetic. II. No other tea gives any energy at all.

Q2. Statement: The college has made 75% attendance compulsory for appearing in the final examination. Conclusions: I. Students with less than 75% attendance will not be allowed to appear for the final examination. II. Every student with 75% or more attendance will pass the final examination.

Q3. Passage: In Shivapur village, most farmers grow soybean in the kharif season. This year the monsoon arrived three weeks late. Because of the delay, the first sowing failed for many farmers, and they had to buy fresh seed and sow a second time. Which of the following can be inferred from the passage?

Q4. Statement: The municipal corporation has added 50 new buses on the city’s busiest routes. Assumptions: I. The new buses will completely end traffic jams in the city. II. All people in the city travel only by bus.

↑ Back to topic list

Reasoning · Verbal reasoning · Topic 11 of 14

Analogy & Classification (Odd One Out)

Mark as done

Think of it like…

The shelves of a kirana shop

Walk into your neighbourhood kirana shop. Kaka has arranged the shelves by a rule: grains on one rack (rice, wheat, jowar, bajra), oils on another, soaps and detergents far away from food. If someone keeps a packet of washing powder next to the jowar, you notice at once — it breaks the shelf rule. Kaka also sells by a pattern: sugar by the kilo, oil by the litre, eggs by the dozen. Ask “Sugar is to kilo as oil is to what?” and the answer is litre, because the relationship (item → unit it is sold in) must be the same on both sides. Analogy and odd-one-out questions test whether you can read Kaka’s rule.

In everyday life	In the exam topic
The label on a shelf	The common property shared by the group
Washing powder kept next to jowar	The odd one out: breaks the common property
Sugar → kilo, oil → litre	Analogy A : B :: C : D — same relationship on both sides
Checking every item on the rack	Test your rule on every option before choosing

The concept

In an analogy (A : B :: C : ?), first say the relationship between A and B in one sentence: “A pen is used to write.” Then apply exactly that sentence to C: “A knife is used to … cut.” Common relationships: tool–use, worker–workplace, item–unit, part–whole, cause–effect, synonym, antonym, animal–young one, and number rules (n², n³, n² + 1, ×2 + 1).

In classification (odd one out), find the property that all but one option share. For numbers, test: prime or not, even/odd, perfect square or cube, divisible by some number, sum of digits. For letter groups, convert to positions and check gaps (consecutive letters, +2 gaps, vowels). For words, check category (grains, fruits, instruments), and then finer details if two candidates appear.

If more than one rule seems to work, pick the most specific and complete one — the rule that makes exactly one option stand out. A rule that makes two options odd is the wrong rule.

Key points & formulas

Say the A–B relationship as a sentence, then reuse the sentence for C
Common word relations: tool–use, worker–workplace, item–unit, part–whole, synonym/antonym
Number analogies: test n², n³, n² ± 1, n³ ± n, n × (n + 1)
Number odd-one-out: prime, square, cube, even/odd, digit sum
Letter groups: convert to positions and compare gaps
A good rule leaves exactly one item out
Keep the order: A : B :: C : D, not B : A

Worked example

Find the odd one out: 27, 64, 125, 144, 216

Show solution

Look for a shared property. 27 = 3³, 64 = 4³, 125 = 5³, 216 = 6³ — all cubes.
144 = 12² is a perfect square but not a perfect cube.
Check that the rule makes only one item odd: yes, only 144 fails “perfect cube”.

Answer: 144

Exam shortcut

For number analogies, look at the size jump: if 7 becomes about 49, test 7² ± something; if it becomes about 343, test 7³. Then confirm with the second pair before choosing.

Common traps

Reversing the relationship: “Pen : Write” is object → use. Do not answer with the use → object pair.
Choosing a rule that makes two options odd. Go back and look for a more specific property.

Practice · tap an option to check

Q1. Pen : Write :: Knife : ?

Q2. 7 : 50 :: 9 : ?

Q3. Find the odd one out: BCD, FGH, JKL, MNP

Q4. Find the odd one out: Jowar, Bajra, Wheat, Cotton

↑ Back to topic list

Reasoning · Analytical puzzles · Topic 12 of 14

Calendars & Clocks

Mark as done

Think of it like…

The weekly village bazaar

In many Maharashtra villages the weekly bazaar (athavdi bazaar) comes on a fixed day, say Tuesday. If today is bazaar day, then 7 days later, 14 days later, 70 days later — it is Tuesday again. Only the leftover days after full weeks change the weekday: 10 days later is 1 full week + 3 extra days, so it is Friday. A normal year is 52 weeks + 1 extra day, so next year the bazaar date falls one weekday later; a leap year has 2 extra days. Clocks work like two bullock carts going round the same circular track at a sugarcane crusher: the minute-hand cart runs 6° per minute, the hour-hand cart only 0.5° per minute, so the fast one gains 5.5° every minute.

In everyday life	In the exam topic
Bazaar day returns every 7 days	Weekdays repeat in a cycle of 7
Leftover days after full weeks	Odd days = number of days mod 7
Normal year = 52 weeks + 1 day; leap year + 2 days	Year shift: +1 weekday (ordinary), +2 (leap)
Fast cart gains on the slow cart every minute	Minute hand gains 5.5° per minute on the hour hand
Fast cart catching the slow cart	Hands coinciding: M = 60H/11 minutes past H

The concept

Odd days are the days left over after removing complete weeks. Ordinary year = 1 odd day, leap year = 2 odd days. 100 years = 5 odd days, 200 years = 3, 300 years = 1, 400 years = 0. Months: 31-day month = 3 odd days, 30-day = 2, February = 0 (or 1 in a leap year). Count the total odd days up to the date, take mod 7, and read 0 = Sunday, 1 = Monday … 6 = Saturday.

A year is a leap year if divisible by 4, except century years, which must be divisible by 400 (2000 yes, 1900 no). When moving from a date in one year to the same date next year, the day shifts by 2 only if 29 February lies in between. Two years have the same calendar when they start on the same weekday and are both leap or both ordinary.

Clocks: the minute hand moves 6° per minute and the hour hand 0.5° per minute. The angle between them at H:M is |30H − 5.5M|; if this is over 180°, subtract from 360°. The hands coincide at 60H/11 minutes past H, and are together 22 times a day. Mirror-image times are covered in the non-verbal topic.

Key points & formulas

Odd day code: 0 Sun, 1 Mon, 2 Tue, 3 Wed, 4 Thu, 5 Fri, 6 Sat
Ordinary year = 1 odd day; leap year = 2 odd days
100 yrs = 5, 200 yrs = 3, 300 yrs = 1, 400 yrs = 0 odd days
Leap year: ÷ 4, but century years must be ÷ 400
Angle = |30H − 5.5M| (use 360 − angle if > 180°)
Hands coincide at 60H/11 min past H; 22 times a day
Hands at right angles 44 times a day; in a straight line (0° or 180°) 44 times a day

Worked example

What day of the week was 15 August 1947?

Show solution

15 Aug 1947 = 1946 complete years + Jan 1 to Aug 15 of 1947.
1600 years → 0 odd days. 300 years (1601–1900) → 1 odd day.
46 years (1901–1946) have 11 leap years and 35 ordinary: 11 × 2 + 35 = 57 → 57 mod 7 = 1 odd day.
1947 up to 15 Aug: Jan 3 + Feb 0 + Mar 3 + Apr 2 + May 3 + Jun 2 + Jul 3 + 15 days (1) = 17 → 3 odd days.
Total = 0 + 1 + 1 + 3 = 5 → Friday.

Answer: Friday

Exam shortcut

For “same date next year” questions, do not count days. Just ask: does 29 February fall between the two dates? Yes → +2 weekdays; no → +1 weekday.

Common traps

Treating 1900 or 2100 as leap years. Century years are leap only if divisible by 400.
Adding 2 days for a leap year when the period does not actually include 29 February (e.g. 1 March 2024 to 1 March 2025 is only +1).

Practice · tap an option to check

Q1. If 26 January 2024 was a Friday, what day of the week was 26 January 2025?

Q2. Which of the following years will have the same calendar as 2023?

Q3. What is the angle between the hour hand and the minute hand of a clock at 3:40?

Q4. At what time between 4 and 5 o’clock are the hands of a clock together?

↑ Back to topic list

Reasoning · Logical deduction · Topic 13 of 14

Venn Diagrams & Data Sufficiency

Mark as done

Think of it like…

Sports-day team lists: kabaddi and kho-kho

On school sports day, the PT sir has two lists: the kabaddi team and the kho-kho team. Some students play both. If he simply adds the two lists to order T-shirts, the double players get two T-shirts — so he subtracts the students who appear on both lists. Draw two overlapping circles and it is obvious: only-kabaddi, both, only-kho-kho, and outside the circles are students who play neither. Then the headmaster asks, “Sir, can you tell me how many play only kho-kho?” Sir looks at the note in his hand and says “Yes, this note is enough” or “No, I need the other list too” — he does not need to actually work out the number. That is data sufficiency.

In everyday life	In the exam topic
Kabaddi list and kho-kho list	Two circles in a Venn diagram
Students on both lists	The overlap (intersection), counted once
T-shirts needed = kabaddi + kho-kho − both	n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Students playing neither game	Region outside both circles = Total − n(A ∪ B)
“Is this note enough to answer?”	Data sufficiency: decide if the data is enough, do not finish the sum

The concept

Venn counting: for two groups, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Only A = n(A) − n(A ∩ B). Neither = Total − n(A ∪ B). For three groups, fill the diagram from the centre (all three) outward, subtracting as you go. Venn relationships ask which picture fits three words: if one class is fully part of another (mothers are women), draw it inside; if two classes can share members (women and doctors), draw them overlapping; if they cannot (men and women), keep them apart.

Data sufficiency: a question is followed by two statements. Check statement I alone, then statement II alone, and only if both fail, check them together. You do not need the final value — only whether a single, definite answer is possible. A yes/no question is sufficient if the data always gives the same answer (always yes, or always no).

Standard answers: I alone is sufficient; II alone is sufficient; each alone is sufficient; both together are needed; even both together are not sufficient. Never carry information from statement I into your check of statement II alone.

Key points & formulas

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Only A = n(A) − n(A ∩ B); Neither = Total − n(A ∪ B)
Three sets: n(A ∪ B ∪ C) = A + B + C − (AB + BC + CA) + ABC
Subset (mothers ⊂ women): circle inside; possible overlap: intersecting circles; impossible overlap: separate circles
DS order: I alone → II alone → together
Sufficient = exactly one definite answer (for yes/no: a definite yes or a definite no)
Do not compute the final value in DS — just decide if you could

Worked example

In a class of 50 students, 30 play kabaddi, 25 play kho-kho and 10 play both. How many students play neither game?

Show solution

Students who play at least one game = 30 + 25 − 10 = 45.
Neither = total − at least one = 50 − 45 = 5.
Check with regions: only kabaddi 20, both 10, only kho-kho 15, neither 5 → 20 + 10 + 15 + 5 = 50 ✓.

Answer: 5 students

Exam shortcut

In DS, if two statements each give one equation in the same two unknowns, “together” is usually enough — but first check that the equations are not the same equation written twice.

Common traps

Adding the two groups without subtracting the overlap, so the “both” students are counted twice.
In DS, using a fact from statement I while judging statement II “alone”.

Practice · tap an option to check

Q1. In a group of 60 people, 35 speak Marathi, 30 speak Hindi and 12 speak both. How many speak only Hindi?

Q2. Which description correctly shows the relationship between Mothers, Women and Doctors?

Q3. What is Rahul’s age? I. Rahul is 4 years older than Sohan. II. Sohan’s age is half of Rahul’s age.

Q4. x is a positive integer. Is x even? I. x² is even. II. x + 5 is odd.

↑ Back to topic list

Reasoning · Non-verbal reasoning · Topic 14 of 14

Non-verbal — Mirror/Water Images, Paper Folding, Dice & Figure Series

Mark as done

Think of it like…

The barber shop mirror and the village pond

Sit in the chair at the barber shop and look at the big mirror. The clock on the wall behind you looks strange: the numbers are reversed left-to-right, and when the real time is 4:20 the mirror seems to show 7:40. Your left hand appears on the right, but your head is still on top. Now look at the temple reflected in the village pond: the top of the shikhar is at the bottom of the water, but left and right are unchanged. Fold a paper to cut a kite or a rangoli stencil and every cut you make appears on both sides of the fold when you open it. And in a game of Ludo, the dice never shows two opposite faces together. Non-verbal reasoning is just these everyday flips, folds and rolls, done in your head.

In everyday life	In the exam topic
Barber shop wall mirror	Mirror image: left ↔ right, top and bottom unchanged
Clock seen in the barber’s mirror	Mirror time = 11:60 − actual time (12:00 − time)
Temple reflected in the pond	Water image: top ↔ bottom, left and right unchanged
Cutting a folded rangoli stencil	Paper folding: every cut is copied across each fold
Ludo dice never shows two opposite faces together	Faces seen together are adjacent; the missing one is opposite

The concept

Mirror image (mirror placed to the left or right): left and right swap, the order of letters in a word reverses, and each letter is flipped. Letters that look the same in a vertical mirror: A, H, I, M, O, T, U, V, W, X, Y. Water image (mirror below): top and bottom swap. Letters unchanged in water: B, C, D, E, H, I, K, O, X. For a clock, mirror time = 11:60 − actual time (if the time is after 11:00, use 23:60 − time).

Paper folding: each fold doubles the number of layers. A single punch through k layers makes k holes when unfolded (if the punch is not on a fold line). A cut on the folded edge opens into a shape that is symmetric about the fold — half a circle cut on a fold becomes a full circle.

Dice: the three faces seen in one view are all adjacent to each other, so none of them is opposite another. Collect all faces adjacent to a number; the one left out is opposite it. On a standard dice, opposite faces add up to 7 (1–6, 2–5, 3–4). Figure series: look for a fixed rotation (45° or 90° clockwise), an element added or removed each step, or a shape moving one corner at a time.

Key points & formulas

Mirror (vertical): left ↔ right; water (horizontal): top ↔ bottom
Mirror clock time = 11:60 − actual time
Vertical-mirror-symmetric letters: A H I M O T U V W X Y
Water-symmetric letters: B C D E H I K O X
n folds → 2ⁿ layers → 2ⁿ holes from one punch (not on a fold)
Faces seen together on a dice are adjacent, never opposite
Standard dice: opposite faces sum to 7
Figure series: check rotation angle and direction first

Worked example

A wall clock shows 4:20. What time does its image in a plane mirror show?

Show solution

For a mirror image of a clock, subtract the time from 11:60 (which is the same as 12:00).
11:60 − 4:20 → hours 11 − 4 = 7, minutes 60 − 20 = 40.
Check: real 4:20 has the hour hand just past 4 (right side); the mirror moves it to between 7 and 8 (left side), and the minute hand from 4 to 8 — that is 7:40.

Answer: 7:40

Exam shortcut

For mirror clocks, just make the two times add up to 12:00. Real 2:35 + mirror ? = 12:00 → mirror 9:25. You can verify any option in two seconds this way.

Common traps

Mixing up mirror and water images: a mirror on the side flips left–right, water below flips top–bottom.
In dice questions, assuming a standard dice (opposite faces sum to 7) when the views show it is not standard. Always use the views first.

Practice · tap an option to check

Q1. A mirror is held vertically to the right of each word. Which word looks exactly the same in the mirror?

Q2. Two views of the same dice: in View 1 the visible faces are 2, 3 and 4; in View 2 the visible faces are 2, 5 and 6. Which number is opposite 2?

Q3. A square sheet of paper is folded once along its vertical centre line. A semicircle is cut out from the middle of the folded edge. What will you see when the paper is opened?

Q4. An arrow changes in a fixed way: ↑, ↗, →, ↘, ↓, ? What comes next?

↑ Back to topic list