Puzzles

Prerequisites

Series and Analogies — the habit of finding a rule and checking it on every item.
Simple equations in one unknown, from Basic Numeracy.
Seating arrangements, blood relations and direction sense are covered separately in Logical Reasoning; this page covers the other puzzle families.

The Hook: The Posting List

Connect

A clerk at the district office has four new officers to post — but the order sheet has been torn and only some lines survive: “Bharti goes to Korba.” “The forest officer goes to Bastar.” “Anil is posted neither to Bastar nor to Durg.” “Deepa handles health.” “Whoever goes to Surguja handles revenue.”

Can the clerk rebuild the full list from these fragments? Yes — and the method is exactly what a CSAT grid puzzle tests.

The question this page answers: how do you turn a set of scattered clues into one certain answer, quickly, without guessing?

Why it matters for CGPSC: puzzles (पहेली) reward a method, not knowledge. With a table drawn the right way, a puzzle that looks long takes two minutes and usually answers two or three questions at once. Without a table, it is the fastest way to lose time in Prelims Paper II.

Ranking and Position

For a single row (or a class ranked from top to bottom):

$$\boxed{\text{Total} = \text{Position from left} + \text{Position from right} - 1}$$

The person is counted in both positions, hence the “−1”.

Find	Formula	Example
Total in the row	$ L + R - 1 $	14th from top, 23rd from bottom → $ 14 + 23 - 1 = 36 $
Rank from the other end	$ \text{Total} - \text{Rank} + 1 $	12th from top in a class of 40 → $ 40 - 12 + 1 = 29 $th from bottom
People between A and B (no overlap)	$ \text{Total} - (L_A + R_B) $	50 in a row, A 12th from left, B 20th from right → $ 50 - 32 = 18 $

Worked example: interchange

In a row, A is 10th from the left and B is 15th from the right. They interchange places, and A becomes 20th from the left. How many people are in the row?

After the swap, A stands exactly where B stood. That spot is 20th from the left (A’s new position) and 15th from the right (B’s old position). Total $ = 20 + 15 - 1 = 34 $.

Check for overlap

For “how many between”, first convert B’s position to “from the left”: $ 50 - 20 + 1 = 31 $. Between 12th and 31st there are $ 31 - 12 - 1 = 18 $ people. If B’s left-position comes out smaller than A’s, the two counts overlap and the formula changes — always convert to one end first.

Ordering by Comparison

Clues like “P is taller than Q” build a chain. Write the chain on one line, tallest on the left, and insert each new clue into it.

Worked example

P is taller than Q but shorter than R. S is shorter than Q. T is taller than R. Who is the tallest, and who is in the middle?

“P taller than Q, shorter than R”: R > P > Q
“S shorter than Q”: R > P > Q > S
“T taller than R”: T > R > P > Q > S

Tallest: T. Middle (3rd of 5): P.

When the chain is incomplete

If a clue only says “U is shorter than R”, U could be anywhere below R. Then “Who is the shortest?” cannot be determined — and CSAT does include that as an option. Do not force a position the clues do not give.

Grid (Distribution) Puzzles

A grid puzzle matches several people with several attributes (district, department, day, colour). The method:

Draw the table: people as rows, one column per attribute.
Fill the direct clues first (“Bharti goes to Korba”).
Apply the negative clues (“Anil is not in Bastar or Durg”) — they often leave only one option.
Use the linking clues (“the forest officer is in Bastar”) once some cells are fixed.
Check every clue against the finished table.

Worked example: the posting list (hypothetical)

Four officers — Anil, Bharti, Chetan and Deepa — are posted to four districts (Bastar, Durg, Korba, Surguja) and four departments (Revenue, Education, Health, Forest), one each.

Bharti is posted to Korba.
Anil is posted neither to Bastar nor to Durg.
The Forest officer is posted to Bastar.
Deepa works in Health.
The officer in Surguja works in Revenue.

Solution. From clue 1, Korba = Bharti. Clue 2 leaves Anil only Surguja, so by clue 5 Anil = Revenue. Chetan and Deepa share Bastar and Durg. Bastar’s officer is in Forest (clue 3), but Deepa is in Health (clue 4), so Chetan = Bastar, Forest and Deepa = Durg, Health. The only department left, Education, goes to Bharti.

Officer	District	Department
Anil	Surguja	Revenue
Bharti	Korba	Education
Chetan	Bastar	Forest
Deepa	Durg	Health

Counting Puzzles

These are short “think, don’t list” puzzles with fixed formulas:

Puzzle	Formula	Example
Handshakes among $ n $ people (each pair once)	$ \frac{n(n-1)}{2} $	10 people → 45
Round-robin league (each team plays every other once)	$ \frac{n(n-1)}{2} $	8 teams → 28 matches
Knock-out tournament	$ n - 1 $ matches	64 players → 63 matches
Heads and legs (hens and cows)	Cows $ = \frac{\text{Legs} - 2\times\text{Heads}}{2} $	40 heads, 110 legs → 15 cows, 25 hens

Why a knock-out needs n − 1 matches

Every match eliminates exactly one player, and all but one player (the champion) must be eliminated. No bracket drawing needed.

Painted Cubes

A large cube is painted on all six faces and cut into $ n\times n\times n $ equal small cubes.

Small cubes with	Where they are	Count
3 faces painted	Corners	$ 8 $ (always)
2 faces painted	Edges (not corners)	$ 12(n-2) $
1 face painted	Faces (not edges)	$ 6(n-2)^2 $
No face painted	Hidden inside	$ (n-2)^3 $
Total		$ n^3 $

$ n $	3 faces	2 faces	1 face	0 faces	Total
3	8	12	6	1	27
4	8	24	24	8	64
5	8	36	54	27	125

The four counts always add up to $ n^3 $ — use that as a check.

Memory Tricks

Anchor these

“Both ends, minus one” — Total = left + right − 1, because the person is counted twice.
Painted cube: “8, 12, 6, 1” with powers 0, 1, 2, 3 — a cube has 8 corners, 12 edges and 6 faces, and 1 hidden core. Multiply each by $ (n-2)^0, (n-2)^1, (n-2)^2, (n-2)^3 $ to get the 3-, 2-, 1- and 0-face counts.
Knock-out = n − 1; league = handshakes.
Grid puzzles: “Direct, Negative, Link, Check” — the order in which to use the clues.

Don’t Confuse

This	vs	That
Total in row $ = L + R - 1 $		People between $ = \text{Total} - (L_A + R_B) $
Round-robin: $ \frac{n(n-1)}{2} $ matches		Knock-out: $ n - 1 $ matches
2 faces painted: $ 12(n-2) $		1 face painted: $ 6(n-2)^2 $
“Cannot be determined” — clues are incomplete		“None of these” — clues are complete but no option matches

Common Mistakes

Trap 1: Forgetting the −1

14th from the top and 23rd from the bottom is 36, not 37. Options always include the “just add” answer.

Trap 2: Solving a grid puzzle in your head

Holding four people and two attributes in memory leads to a contradiction three clues later. A 30-second table saves minutes and serves every question in the set.

Trap 3: Using n instead of n − 2 for painted cubes

The inner cube and the edge pieces exclude the corners, so the side length to use is $ n - 2 $. For $ n = 4 $, unpainted cubes are $ 2^3 = 8 $, not $ 4^3 $ or $ 3^3 $.

Practice

Level 1: Foundation

In a row, Seema is 9th from the left and 12th from the right. How many people are in the row?
How many handshakes take place among 6 people if each pair shakes hands once?
A 3 × 3 × 3 painted cube is cut into 27 small cubes. How many have no face painted?
How many matches are played in a knock-out tournament of 16 teams?

Answers

$ 9 + 12 - 1 = 20 $.
$ \frac{6\times 5}{2} = 15 $.
$ (3-2)^3 = 1 $ — only the centre cube.
$ 16 - 1 = 15 $.

Level 2: Prelims MCQs

Q1. In a class of 45 students, Ravi’s rank is 17th from the top. What is his rank from the bottom?

(a) 28 (b) 29 (c) 30 (d) 27

Answer: (b) 29

Rank from the bottom $ = 45 - 17 + 1 = 29 $. Option (a) forgets the “+1”.

Q2. In a row, A is 8th from the left and B is 12th from the right. When they interchange their positions, A becomes 15th from the left. How many people are there in the row?

(a) 25 (b) 27 (c) 26 (d) 24

Answer: (c) 26

After the interchange, A occupies B’s old place, which is 15th from the left and 12th from the right. Total $ = 15 + 12 - 1 = 26 $.

Q3. A cube painted on all faces is cut into 125 equal small cubes. How many small cubes have exactly two faces painted?

(a) 54 (b) 27 (c) 48 (d) 36

Answer: (d) 36

$ 125 = 5^3 $, so $ n = 5 $. Two faces painted $ = 12(n-2) = 12\times 3 = 36 $. Option (a) 54 is the one-face count $ 6(n-2)^2 $, and (b) 27 is the unpainted count $ (n-2)^3 $.

Q4. Among five friends P, Q, R, S and T: Q is heavier than R, P is lighter than R, S is heavier than Q, and T is lighter than P. Consider the following statements:

S is the heaviest.
R is third in order of weight.
T is heavier than P.

Which of the statements given above is/are correct?

(a) 1 only (b) 1 and 2 only (c) 2 and 3 only (d) 1, 2 and 3

Answer: (b) 1 and 2 only

The chain is S > Q > R > P > T. S is the heaviest (1 correct) and R is third (2 correct). T is the lightest, so statement 3 is wrong.

Q5. Match List I with List II:

List I	List II
A. Handshakes among 12 people (each pair once)	1. 20
B. Matches in a knock-out tournament of 32 teams	2. 28
C. Matches in a round-robin league of 8 teams	3. 31
D. Cows among 30 animals (hens and cows) with 100 legs in all	4. 66

(a) A-4, B-3, C-2, D-1 (b) A-4, B-2, C-3, D-1 (c) A-3, B-4, C-2, D-1 (d) A-4, B-3, C-1, D-2

Answer: (a) A-4, B-3, C-2, D-1

A: $ \frac{12\times 11}{2} = 66 $. B: $ 32 - 1 = 31 $. C: $ \frac{8\times 7}{2} = 28 $. D: cows $ = \frac{100 - 2\times 30}{2} = 20 $ (and 10 hens; check $ 20\times 4 + 10\times 2 = 100 $).

Level 3: Speed Drill — Solve in 60 Seconds

Each item should take under 60 seconds with the formulas on this page. Puzzles in the reasoning part of Mains Paper IV (GS-II) also expect the table or chain to be shown.

Kiran is 14th from the top and 23rd from the bottom in a class. Class strength?
50 people in a row; A is 12th from the left and B is 20th from the right. How many people are between them?
A farmyard has 40 heads and 110 legs (hens and cows only). How many cows?
A 4 × 4 × 4 painted cube is cut into unit cubes. How many have no face painted?
10 people each shake hands once with every other person. Total handshakes?
A knock-out chess tournament has 64 players. How many matches?

Answers with the formula used

36 — $ 14 + 23 - 1 $.
18 — $ 50 - (12 + 20) $.
15 — $ \frac{110 - 80}{2} $.
8 — $ (4-2)^3 $.
45 — $ \frac{10\times 9}{2} $.
63 — $ n - 1 $.

Quick Revision Table

Idea	Formula / Fact
Total in a row	Left + Right − 1
Rank from other end	Total − Rank + 1
Interchange questions	New position of one = old position of the other
Comparison clues	Build one chain; “cannot be determined” if a link is missing
Grid puzzles	Direct → Negative → Link → Check
Handshakes / league	$ \frac{n(n-1)}{2} $
Knock-out	$ n - 1 $
Painted cube	8 · $ 12(n-2) $ · $ 6(n-2)^2 $ · $ (n-2)^3 $

Key Takeaways

Total = left + right − 1; for “between” questions, convert both positions to the same end first.
Comparison clues form a chain — write it on one line and add each clue to it.
Grid puzzles are solved by a table: direct clues, then negative clues, then linking clues, then a full check.
Counting puzzles have fixed formulas: handshakes and leagues $ \frac{n(n-1)}{2} $, knock-outs $ n - 1 $.
Painted-cube counts use $ n - 2 $ and always add to $ n^3 $.

What’s Next

Calendars — the odd-days method for any date.
Clocks — angles, overlaps and faulty clocks.
Logical Reasoning — seating arrangements, blood relations and direction sense.
Decision Making — the other CSAT block that rewards a fixed method.
General Mental Ability — Quick Revision — every formula from this chapter on one page.

This	vs	That
Total in row \( = L + R - 1 \)		People between \( = \text{Total} - (L_A + R_B) \)
Round-robin: \( \frac{n(n-1)}{2} \) matches		Knock-out: \( n - 1 \) matches
2 faces painted: \( 12(n-2) \)		1 face painted: \( 6(n-2)^2 \)
“Cannot be determined” — clues are incomplete		“None of these” — clues are complete but no option matches