UTKALPRESS
Puzzle
Arithmetic & Reasoning Book Note • Concepts • Formulae • Shortcuts
1. What is a Puzzle?
A puzzle is a logical reasoning problem in which several
facts, conditions or clues are given and we have to determine the correct
arrangement, order, relationship or missing information.
Puzzle questions test logical thinking, arrangement ability, comparison, deduction and systematic elimination.
Puzzle questions test logical thinking, arrangement ability, comparison, deduction and systematic elimination.
2. Common Types of Puzzles
| Type | Main Idea |
|---|---|
| Linear Arrangement | People or objects arranged in a row. |
| Circular Arrangement | People arranged around a circle. |
| Floor Puzzle | People placed on different floors of a building. |
| Scheduling Puzzle | People/events assigned to different days or months. |
| Box/Floor Puzzle | Objects or people assigned to positions. |
| Ranking Puzzle | Determine positions from top/bottom or left/right. |
| Grouping Puzzle | People or objects divided into groups. |
| Selection Puzzle | Select members according to given conditions. |
3. Basic Puzzle Strategy
Step 1: Read all clues carefully.
Step 2: Identify the fixed information.
Step 3: Draw a table, line or circle.
Step 4: Place definite information first.
Step 5: Use negative conditions to eliminate positions.
Step 6: Apply remaining clues systematically.
Step 7: Verify the complete arrangement against every clue.
Step 2: Identify the fixed information.
Step 3: Draw a table, line or circle.
Step 4: Place definite information first.
Step 5: Use negative conditions to eliminate positions.
Step 6: Apply remaining clues systematically.
Step 7: Verify the complete arrangement against every clue.
4. Linear Arrangement
Important Position Formula:
Position from right = Total persons − Position from left + 1
Example: There are 10 students in a row. Ravi is
4th from the left. What is his position from the right?
Position from right = 10 − 4 + 1
= 7th
Position from right = 10 − 4 + 1
= 7th
Combined Position Formula:
If a person is
If a person is
ath from left and bth from right:
Total persons = a + b − 1
Example: Sita is 8th from the left and 13th from
the right.
Total = 8 + 13 − 1 = 20
Total = 8 + 13 − 1 = 20
5. Immediate Neighbour
If A sits immediately to the left of B, the order is:
A B.
If A sits immediately to the right of B, the order is: B A.
"Immediately" means no person/object is present between them.
If A sits immediately to the right of B, the order is: B A.
"Immediately" means no person/object is present between them.
UTKALPRESS
Puzzle
Advanced Arrangement & Position Concepts
6. Ranking and Position
Total persons:
Persons between two positions:
Left position + Right position − 1
Persons between two positions:
|Position A − Position B| − 1
Example: A is 7th and B is 15th in a row.
Persons between them = 15 − 7 − 1
= 7 persons
Persons between them = 15 − 7 − 1
= 7 persons
7. Ranking with Equal Positions
If two people have the same position from opposite ends, the
total number can be found directly.
Example: A person is 12th from the top and 19th
from the bottom.
Total = 12 + 19 − 1
= 30
Total = 12 + 19 − 1
= 30
8. Circular Arrangement
In a circular puzzle, there is no absolute left or right until a
reference direction is established.
For a clockwise arrangement: clockwise = movement in the direction of a clock's hands.
Anticlockwise = opposite direction.
For a clockwise arrangement: clockwise = movement in the direction of a clock's hands.
Anticlockwise = opposite direction.
Number of arrangements of n different people around a circle:
This applies when arrangements that differ only by rotation are considered identical.
(n − 1)!
This applies when arrangements that differ only by rotation are considered identical.
Example: In how many circular arrangements can
5 people sit?
= (5 − 1)!
= 4!
= 24
= (5 − 1)!
= 4!
= 24
9. Fixed Position in Circular Puzzle
Fix one person at a reference position first. Then arrange the
remaining people around the circle. This removes unnecessary
rotational duplication.
Example: Six people A, B, C, D, E and F sit around
a circular table. If A is fixed, the remaining five can be arranged
in:
5! = 120 ways.
5! = 120 ways.
10. Clockwise and Anticlockwise Clues
If A is immediately clockwise from B, write:
B → A.
If A is immediately anticlockwise from B, write: A → B.
Always keep the same direction convention throughout the puzzle.
If A is immediately anticlockwise from B, write: A → B.
Always keep the same direction convention throughout the puzzle.
11. Floor Puzzles
Technique: Draw floors vertically and number them
from bottom to top. Place fixed floors first, then apply clues such
as "above", "below", "immediately above" and "not on floor".
Example: A lives immediately above B. If B is on
the 3rd floor, A must be on the 4th floor.
Important: "Above" does not always mean immediately
above. "Immediately above" means exactly one floor higher.
UTKALPRESS
Puzzle
Scheduling, Grouping & Selection
12. Day and Month Scheduling
Scheduling puzzles assign people, events, subjects or activities
to different days, dates or months according to a set of conditions.
Best method:
1. Write all available days/months in a row.
2. Mark fixed assignments.
3. Cross out prohibited positions.
4. Apply "before", "after", "immediately before" and "between" conditions.
1. Write all available days/months in a row.
2. Mark fixed assignments.
3. Cross out prohibited positions.
4. Apply "before", "after", "immediately before" and "between" conditions.
13. Important Scheduling Rules
| Clue | Meaning |
|---|---|
| A before B | A occurs earlier than B. |
| A immediately before B | A and B occupy consecutive positions. |
| A after B | A occurs later than B. |
| A between B and C | B and C are on opposite sides of A. |
| A not on Monday | Monday is eliminated for A. |
Example: Five seminars are held from Monday to
Friday. Mathematics is immediately before Science. If Mathematics
is on Tuesday, Science must be on Wednesday.
14. Grouping Puzzles
In grouping puzzles, people or objects are divided into teams,
committees or groups while satisfying conditions such as:
together, separate, exactly one, at least one, not both,
or one of the two.
Combination Formula:
Number of ways to choose r objects from n objects:
Number of ways to choose r objects from n objects:
nCr = n! / [r!(n−r)!]
Example: In how many ways can 3 people be selected
from 5 people?
Answer = 10 ways
5C3 = 5!/(3!2!) = 10
Answer = 10 ways
15. Selection with Restrictions
When a condition says two people must be together,
treat them as one unit wherever appropriate.
When two people cannot be together, first count all possibilities and subtract the unwanted cases.
When two people cannot be together, first count all possibilities and subtract the unwanted cases.
Complement Method:
Required cases = Total cases − Unwanted cases
Example: From 6 people, select 2 people. How many
selections are possible?
Total =
If A and B cannot be selected together, remove the one unwanted selection AB:
Required = 15 − 1 = 14
Total =
6C2 = 15
If A and B cannot be selected together, remove the one unwanted selection AB:
Required = 15 − 1 = 14
16. "Either / Or" and "Both" Conditions
Both A and B: Both conditions must be satisfied.
Either A or B: Usually means at least one of the two conditions, unless the question specifically says "but not both".
Neither A nor B: Both A and B are excluded.
Either A or B: Usually means at least one of the two conditions, unless the question specifically says "but not both".
Neither A nor B: Both A and B are excluded.
Always read whether "either...or" is being used inclusively
(one or both) or exclusively (exactly one). The wording of the
question determines the interpretation.
17. Useful Counting Formulae
| Situation | Formula |
|---|---|
| Arrange n objects | n! |
| Select r from n | nCr |
| Arrange r from n | nPr = n!/(n−r)! |
| Circular arrangement | (n−1)! |
| Complement counting | Total − unwanted |
UTKALPRESS
Puzzle
Shortcuts • Solved Examples • Practice
18. Master Shortcut Table
| Concept | Formula / Shortcut |
|---|---|
| Right position | Total − Left + 1 |
| Total persons | Left + Right − 1 |
| Persons between | |A − B| − 1 |
| Linear arrangement | n! for n distinct objects |
| Circular arrangement | (n−1)! |
| Selection | nCr |
| Arrangement | nPr |
| Restricted cases | Total − unwanted cases |
| Weighted average | Σwx / Σw |
19. Solved Puzzle Example — Linear
Question: Six people A, B, C, D, E and F sit in a
row. A sits at the extreme left. B sits immediately to the right
of A. C sits at the extreme right. D sits immediately to the left
of C. Who can occupy the remaining positions?
``` Positions:
1 = A
2 = B
5 = D
6 = C
Remaining positions = 3 and 4. Therefore E and F occupy positions 3 and 4 in either order. ```
``` Positions:
1 = A
2 = B
5 = D
6 = C
Remaining positions = 3 and 4. Therefore E and F occupy positions 3 and 4 in either order. ```
20. Solved Puzzle Example — Ranking
Question: In a class, Mohan is 9th from the top
and 17th from the bottom. Find the total number of students.
``` Total = 9 + 17 − 1
= 25 students. ```
``` Total = 9 + 17 − 1
= 25 students. ```
21. Solved Puzzle Example — Circular
Question: Eight friends sit around a circular table.
How many different arrangements are possible if rotations are
considered identical?
``` Number of arrangements = (8 − 1)!
= 7!
= 5040. ```
``` Number of arrangements = (8 − 1)!
= 7!
= 5040. ```
22. Puzzle-Solving Checklist
✓ Identify the number of positions.
✓ Mark fixed positions first.
✓ Translate every clue into a simple condition.
✓ Use symbols such as →, ←, ↑, ↓ and × where useful.
✓ Keep separate possibilities in separate rows.
✓ Do not assume information that is not given.
✓ Check every clue before selecting the answer.
✓ Mark fixed positions first.
✓ Translate every clue into a simple condition.
✓ Use symbols such as →, ←, ↑, ↓ and × where useful.
✓ Keep separate possibilities in separate rows.
✓ Do not assume information that is not given.
✓ Check every clue before selecting the answer.
23. Practice Questions
1. A person is 8th from the left in a row of
25 people. What is his position from the right?
2. A student is 12th from the top and 18th from
the bottom. Find the total number of students.
3. A is 6th and B is 14th in a row. How many
people are between A and B?
4. In how many ways can 6 people be arranged in
a straight line?
5. In how many ways can 6 people sit around a
circular table?
6. In how many ways can 2 people be selected from
7 people?
7. In how many ways can 3 people be selected from
8 people?
8. Five books are arranged on a shelf. How many
arrangements are possible?
9. If A is immediately before B and B is immediately
before C, write the order of the three persons.
10. Seven persons are standing in a row. Rahul is
3rd from the left. What is his position from the right?
24. Answer Key
| Q. | Answer | Q. | Answer |
|---|---|---|---|
| 1 | 18th | 6 | 21 |
| 2 | 29 | 7 | 56 |
| 3 | 7 | 8 | 120 |
| 4 | 720 | 9 | A-B-C |
| 5 | 120 | 10 | 5th |
Final Note: Puzzle questions become easier when
the verbal clues are converted into a visual arrangement.
Do not try to solve every clue mentally. A clear table, diagram
or position chart reduces mistakes and makes the deduction process
much faster.
