Venn, Clocks & Puzzles
Sort groups with circles, read clocks and calendars like a detective, and crack seating, family and logic puzzles step by step.
1Venn diagrams: how groups are related
A Venn diagram uses circles to show groups (classes). How the circles sit — inside, overlapping or apart — tells us how the groups are related.
- Inside: every member of the small group belongs to the big group (all dogs are animals).
- Overlapping: some members are in both groups, some are not (some singers are dancers).
- Separate: no member is in both groups (no cat is a dog).
Choosing the right diagram for three groups
Take the groups two at a time and decide inside / overlapping / separate for each pair. Then draw.
Which diagram best shows Mothers, Women and Doctors?
- Mothers and Women: every mother is a woman → Mothers circle inside Women.
- Women and Doctors: some women are doctors, some doctors are men → overlapping.
- Mothers and Doctors: some mothers are doctors → overlapping. So the Doctors circle must cut across both circles.
Answer: Mothers inside Women, with Doctors overlapping both (as in the picture).
| Groups | Diagram |
|---|---|
| Tea, Coffee, Beverages | two separate circles inside one big circle |
| Delhi, India, Asia | one inside another inside another |
| Pen, Car, Chair | three separate circles |
| Teachers, Men, Women | Men and Women separate; Teachers overlaps both |
| Fruits, Mangoes, Vegetables | Mangoes inside Fruits; Vegetables separate |
“Doctors” and “Women” are not separate — many women are doctors. Don't think of the usual picture in your head; ask “can someone be in both groups?” If yes, the circles must overlap.
Which diagram best shows Fruits, Mangoes and Vegetables?
2Reading numbers from Venn diagrams
Use the diagram. (a) How many play cricket? (b) How many play only one game? (c) How many children are in the class?
- (a) Cricket circle = only cricket + both = 12 + 5 = 17.
- (b) Only one game = only cricket + only football = 12 + 8 = 20.
- (c) Everyone = 12 + 5 + 8 + 3 (neither) = 28.
Answer: (a) 17 (b) 20 (c) 28
From the three-circle diagram, find: (a) how many like Hindi, (b) how many like exactly two subjects, (c) how many like English and Maths but not Hindi.
- (a) Add every number inside the Hindi circle: 10 + 4 + 5 + 2 = 21.
- (b) Exactly two = the three regions where just two circles overlap: 4 + 3 + 5 = 12. (The 2 in the middle like all three, not exactly two.)
- (c) English and Maths but not Hindi = the region inside English and Maths, outside Hindi = 3.
Answer: (a) 21 (b) 12 (c) 3
“Likes Hindi” means everyone inside the Hindi circle, including the overlaps — not just the “only Hindi” number. And “exactly two” leaves out the middle region.
In the cricket–football diagram, how many children play at least one of the two games?
3Clock reasoning
A clock face is a full turn of 360° divided into 12 equal gaps, so each gap between two numbers is 360° ÷ 12 = 30°.
- The minute hand goes all the way round (360°) in 60 minutes → 6° every minute.
- The hour hand moves only one gap (30°) in 60 minutes → ½° every minute. So at 3:30 it is halfway between 3 and 4, not on 3.
What is the angle between the hands at 4:00? At 6:00?
- At 4:00 the minute hand is on 12 and the hour hand on 4 → 4 gaps × 30° = 120°.
- At 6:00 → 6 gaps × 30° = 180° (a straight line).
Answer: 120° and 180°
What is the angle between the hands at 9:30?
- Minute hand on 6.
- Hour hand halfway between 9 and 10.
- From 6 to 9 is 3 gaps; plus the half gap to the hour hand = 3½ gaps.
- 3½ × 30° = 105°.
Answer: 105°
| Time | Angle | Shape |
|---|---|---|
| 12:00 | 0° | hands together |
| 3:00 and 9:00 | 90° | right angle |
| 6:00 | 180° | straight line |
| 2:00 and 10:00 | 60° | — |
| 6:30 | 15° | hour hand halfway between 6 and 7 |
| 12:30 | 165° | hour hand halfway between 12 and 1 |
In one day (24 hours) the two hands are together 22 times, at right angles 44 times and in a straight line (opposite) 22 times.
Clocks that gain or lose time
A clock gains 3 minutes every hour. It is set right at 9:00 AM. What time will it show at 1:00 PM?
- From 9:00 AM to 1:00 PM is 4 hours.
- It gains 4 × 3 = 12 minutes.
- A clock that gains runs ahead: 1:00 PM + 12 min.
Answer: 1:12 PM
A watch loses 2 minutes every hour. It is set right at 6:00 AM. What will it show at 6:00 PM the same day?
- 6 AM to 6 PM = 12 hours.
- It loses 12 × 2 = 24 minutes.
- A watch that loses falls behind: 6:00 PM − 24 min.
Answer: 5:36 PM
At 3:30 many students say 90° because they put the hour hand on 3. It has moved on! Always ask: where is the hour hand after the minutes have passed?
What is the angle between the hands of a clock at 5:00?
4Calendar reasoning
Days of the week repeat every 7 days. So to move forward many days, only the remainder after dividing by 7 matters.
Today is Monday. What day will it be after 30 days?
- 30 ÷ 7 = 4 weeks and 2 days left over.
- 4 full weeks bring us back to Monday.
- Monday + 2 days = Wednesday.
Answer: Wednesday
The day before yesterday was Thursday. What will be the day after tomorrow?
- Day before yesterday = Thursday → yesterday = Friday → today = Saturday.
- Day after tomorrow = Saturday + 2 = Monday.
Answer: Monday
If 1 March is a Tuesday, what day is 29 March?
- 29 − 1 = 28 days later.
- 28 is exactly 4 weeks, so the day is the same.
Answer: Tuesday
Years and leap years
| Year type | Days | Weeks + extra days | Same date next year moves by |
|---|---|---|---|
| Ordinary year | 365 | 52 weeks + 1 day | 1 day |
| Leap year | 366 (29 Feb) | 52 weeks + 2 days | 2 days (if 29 Feb comes in between) |
- A year is a leap year if it divides exactly by 4 (2024, 2028). But a century year (ending in 00) must divide by 400: 2000 was a leap year, 1900 was not.
- Months with 31 days: January, March, May, July, August, October, December. February has 28 days (29 in a leap year); the others have 30.
15 August 2026 is a Saturday. What day is 15 August 2027?
- From 15 Aug 2026 to 15 Aug 2027 there is no 29 February (February 2027 has 28 days).
- So the gap is 365 days = 52 weeks + 1 day.
- Saturday + 1 = Sunday.
Answer: Sunday
Which is the next year that will have exactly the same calendar as 2026?
- A calendar repeats when the extra days add up to a multiple of 7 and both years are of the same type (both ordinary).
- Extra days: 2026 (+1), 2027 (+1), 2028 leap (+2), 2029 (+1), 2030 (+1), 2031 (+1) → total 7. So 2032 starts on the same day as 2026 — but 2032 is a leap year, so its calendar is different. ✘
- Keep going: 2032 (+2), 2033 (+1), 2034 (+1), 2035 (+1), 2036 leap (+2) → total 14, a multiple of 7.
- 2037 is an ordinary year. ✔
Answer: 2037
When counting “after 30 days”, don't count today as day 1. And in “same calendar” questions, starting on the same weekday is not enough — a leap year and an ordinary year never share a calendar.
Today is Wednesday. What day will it be 15 days from today?
5Seating arrangements
Seating questions give you clues about who sits where. Always draw it. Fix the surest clue first, then fill in the rest.
In a row
B, A, C, D and E sit in a row facing North. C is in the middle. A is to the immediate left of C. E is at the right end. D is not at an end. Who is second from the right?
- Draw 5 seats. Facing North, their left is your left.
- C in the middle → seat 3. A immediately left of C → seat 2. E at the right end → seat 5.
- B and D are left for seats 1 and 4. D is not at an end → D in seat 4, B in seat 1.
- Row (left to right): B A C D E. Second from the right = D.
Answer: D
If people sit facing South (towards you), their left is your right. Always check which way they face before using “left” and “right”.
Around a round table
P, Q, R, S, T and U sit around a round table facing the centre. P is opposite S. Q is to the immediate left of P. R is opposite Q. T is to the immediate right of S. Where is U?
- Put P at the top. S is opposite → bottom.
- Facing the centre, left = clockwise. Q is immediately left of P → next seat clockwise from P.
- R is opposite Q.
- Right = anticlockwise. T is immediately right of S → the seat just anticlockwise from S (between Q and S).
- The only empty seat is between R and P.
Answer: U sits between R and P (and opposite T).
Six children sit in a row facing North. Ravi is 2nd from the left and Sam is 2nd from the right. How many children sit between them?
6Blood relations
Blood-relation questions describe a family in words. Turn the words into a family tree, one step at a time.
| Words | Relation |
|---|---|
| Father's or mother's father / mother | Grandfather / grandmother |
| Father's brother | Uncle (chacha / tau) |
| Mother's brother | Uncle (mama) |
| Father's sister / mother's sister | Aunt (bua / mausi) |
| Uncle's or aunt's son or daughter | Cousin |
| Brother's or sister's son / daughter | Nephew / niece |
| Son's wife / daughter's husband | Daughter-in-law / son-in-law |
| Father's or mother's only son (for a girl) | Her brother |
Pointing to a boy, Riya says, “He is the son of my grandfather's only son.” How is the boy related to Riya?
- Riya's grandfather's only son must be Riya's father.
- The son of Riya's father is Riya's brother.
Answer: Brother
A is B's sister. B is C's brother. C is D's daughter. How is A related to D?
- A, B and C are brothers and sisters (children of the same parents).
- C is D's daughter, so D is a parent of all three.
- A is a sister, so A is female.
Answer: A is D's daughter.
X is the father of Y, but Y is not X's son. What is Y to X?
- Y is X's child.
- A child who is not a son is a daughter.
Answer: Daughter
“P is the brother of Q” tells you P is a boy — but it tells you nothing about whether Q is a boy or a girl! Never guess a gender that the question doesn't give.
Meera says, “Rohan is my mother's only brother's son.” How is Rohan related to Meera?
7Logical puzzles and statements
Ordering puzzles
Asha is taller than Bina. Chitra is shorter than Bina. Diya is taller than Asha. Who is the shortest, and who is second tallest?
- Write a line from tallest to shortest and place each clue.
- Asha > Bina; Bina > Chitra → Asha > Bina > Chitra.
- Diya > Asha → Diya > Asha > Bina > Chitra.
Answer: Chitra is the shortest; Asha is second tallest.
Ravi, Sita and Tom have a cat, a dog and a parrot (one each). Ravi does not have the dog or the parrot. Sita does not have the dog. Who has the dog?
- Ravi has neither dog nor parrot → Ravi has the cat.
- Sita does not have the dog, and the cat is taken → Sita has the parrot.
- The dog is left for Tom.
Answer: Tom
For “who has what” puzzles, draw a small grid with names down the side and things across the top. Put ✘ for every “does not”, and ✔ when only one box is left in a row or column.
Statements and conclusions
Treat the statements as true, even if they sound odd. A conclusion follows only if it must be true from the statements — not if it only might be true.
| Statements | Conclusion | Follows? |
|---|---|---|
| All birds have feathers. A parrot is a bird. | A parrot has feathers. | ✔ Yes — must be true |
| Some boys are tall. Rahul is a boy. | Rahul is tall. | ✘ No — he may or may not be |
| All roses are flowers. Some flowers fade quickly. | Some roses fade quickly. | ✘ No — the fast-fading flowers may not be roses |
| All cats are animals. All animals need food. | All cats need food. | ✔ Yes |
Statement: Regular exercise keeps the body fit. Conclusion I: People should exercise regularly. Conclusion II: Only exercise keeps the body fit. Which follows?
- I agrees with the statement: if exercise keeps us fit, it is sensible to exercise → follows.
- II adds the word only. The statement never said nothing else helps (food and sleep matter too) → does not follow.
Answer: Only I follows.
Watch out for strong words that sneak into conclusions: only, all, always, never, must. If the statement did not say it, the conclusion does not follow. Also, “some A are B” never lets you decide about one particular A.
Statements: All pens are blue. Some blue things are costly. Conclusion: Some pens are costly. Does it follow?
8Watch and explore
9Quick recap
- Venn diagrams: inside (all A are B), overlapping (some A are B), separate (no A is B).
- For three groups, decide each pair; e.g. Mothers inside Women, Doctors overlapping both.
- A number in a region counts only that region; a group's total adds every region inside its circle.
- “Exactly two” leaves out the centre; “at least one” is everyone inside the circles.
- Clock: each gap 30°; minute hand 6° per minute; hour hand ½° per minute.
- 3:00 → 90°, 6:00 → 180°, 3:30 → 75°, 9:30 → 105°.
- Gaining clocks run ahead (add), losing clocks fall behind (subtract).
- Days repeat every 7: use the remainder after dividing by 7.
- Ordinary year = 52 weeks + 1 day; leap year = 52 weeks + 2 days; century years need ÷ 400.
- Row facing North: their left = your left. Facing South: reversed.
- Round table facing the centre: left = clockwise, right = anticlockwise.
- Blood relations: draw a family tree; never assume a gender that isn't given.
- A conclusion follows only if it must be true; beware of “only”, “all”, “never”.