Linear Arrangement
Place entities in numbered slots while respecting left/right and adjacency clues.
Note 25
Problem Solving and Puzzle Reasoning
Complete notes for SSC Combined Graduate Level examinations
1The Big Picture — Graphic Mind Map
Circular Seating
Fix one anchor, determine facing, then apply clockwise/anticlockwise relations.
Ranking
Convert positions from both ends and count people between.
Blood Relations
Track generation, gender and connecting links in a family tree.
Direction Sense
Map turns and movement on coordinates; use net displacement.
Calendar & Clock
Reduce repeated cycles: weekdays modulo \(7\), clock positions modulo \(12\).
Age & Order
Translate “older,” “before,” “higher” and “immediately” into inequalities and gaps.
Matrix Puzzle
Compare rows, columns and corresponding cells to find one repeatable rule.
1. Inventory
entities & slots
➜entities & slots
2. Encode
clue symbols
➜clue symbols
3. Anchor
fixed facts
➜fixed facts
4. Link
constraints
➜constraints
5. Audit
every clue
every clue
2The Foundation — Complete Puzzle Toolkit
2.1 What a Puzzle Is
A puzzle gives entities, attributes, slots and constraints. Your model must satisfy all clues simultaneously.
- Entity: person/object/date/place.
- Domain: available values or positions.
- Constraint: relation restricting possibilities.
- Solution: complete assignment satisfying every constraint.
2.2 Universal Workflow
- List entities and categories.
- Draw suitable slots/table/tree/map.
- Place direct/fixed clues.
- Combine linked clues into blocks.
- Branch only when necessary.
- Verify every original clue.
2.3 Clue Strength Hierarchy
| Clue type | Example language | Best representation | Priority |
|---|---|---|---|
| Absolute | first, last, centre, north, Monday | Direct slot | Highest |
| Immediate | immediately before, adjacent, next to | Fixed block | High |
| Relative distance | two places left, three years older | Gap equation | High |
| Ordering | before, taller, ranks above | Chain / inequality | Medium |
| Negative | not next to, not Tuesday | Cross-out | After anchors |
| Either/or | one of \(A,B\), but not both | Cases | Later |
2.4 Linear Arrangement
Number the slots
For \(n\) entities, create positions \(1,2,\ldots,n\). Write a facing arrow because “left” depends on viewpoint.
Blocks
“\(A\) immediately left of \(B\)” forms block \([AB]\). Treat it as one unit until placement.
Gap count
If \(k\) persons sit between \(A\) and \(B\), their position difference is \(k+1\):
\[|p_A-p_B|=k+1\]
2.5 Ranking and Order
Both-end formula
If one person is \(L\)-th from left and \(R\)-th from right:
\[N=L+R-1\]
The person is counted in both ranks, hence subtract one.
Persons between
For known ranks from the same end:
\[\text{between}=|r_1-r_2|-1\]
For opposite-end ranks, first convert one rank using \(r_{\rm right}=N-r_{\rm left}+1\).
2.6 Circular Seating
Break rotational symmetry
Fix one person at the top. Only relative positions matter, so rotating the whole arrangement does not create a new case.
Facing centre
For inward-facing people, left is clockwise and right is anticlockwise from the person's perspective.
Facing outward
Directions reverse: left becomes anticlockwise and right becomes clockwise.
2.7 Blood and Family Relations
Family-tree notation
- Horizontal line: spouses/same generation.
- Vertical line: parent–child.
- Square / circle: male / female, if you choose.
- Move up for parent; down for child.
- Maternal = through mother; paternal = through father.
Generation method
Assign the reference generation \(0\): parents \(+1\), grandparents \(+2\), children \(-1\), grandchildren \(-2\). Then resolve gender and branch.
Relation is directional: “\(A\) is brother of \(B\)” identifies \(A\)'s gender, not \(B\)'s.
2.8 Relation Dictionary
| Target generation | Male | Female | Branch clue |
|---|---|---|---|
| \(+2\) | Grandfather | Grandmother | Maternal/paternal by intermediate parent |
| \(+1\) | Father / uncle / father-in-law | Mother / aunt / mother-in-law | Blood vs marriage path |
| \(0\) | Brother / husband / cousin | Sister / wife / cousin | Sibling, spouse, cousin |
| \(-1\) | Son / nephew / son-in-law | Daughter / niece / daughter-in-law | Child of whom? |
| \(-2\) | Grandson | Granddaughter | Child's child |
2.9 Direction Sense and Route Tracing
Coordinate model
Use East \((+x)\), West \((-x)\), North \((+y)\), South \((-y)\). Add moves component-wise.
Net displacement
\[d=\sqrt{(\Delta x)^2+(\Delta y)^2}\]
Shortest distance ignores the travelled bends; direction comes from signs of \(\Delta x,\Delta y\).
Turns
Right turn \(=90^\circ\) clockwise; left turn \(=90^\circ\) anticlockwise; about-turn \(=180^\circ\). Reduce accumulated turns modulo \(360^\circ\).
2.10 Direction Conversion Table
| Facing | Left turn | Right turn | About-turn |
|---|---|---|---|
| North | West | East | South |
| East | North | South | West |
| South | East | West | North |
| West | South | North | East |
2.11 Calendar Reasoning
Odd days
An ordinary year has \(365=52\times7+1\): one odd day. A leap year has \(366=52\times7+2\): two odd days.
Leap-year rule
A year is leap if divisible by \(4\), except century years must be divisible by \(400\).
Thus \(2000\) is leap; \(1900\) is not.
Weekday shift
\[\text{new day}=\text{old day}+(N\bmod7)\]
Discard complete weeks immediately.
2.12 Month Data Bank
| Days | Months | Odd days |
|---|---|---|
| \(31\) | Jan, Mar, May, Jul, Aug, Oct, Dec | \(3\) |
| \(30\) | Apr, Jun, Sep, Nov | \(2\) |
| \(28\) | Feb in ordinary year | \(0\) |
| \(29\) | Feb in leap year | \(1\) |
2.13 Clock Reasoning
Hand speeds
Minute hand: \(6^\circ\) per minute. Hour hand: \(0.5^\circ\) per minute. Relative speed: \(5.5^\circ\) per minute.
Angle formula
\[\theta=|30h-5.5m|\]
Smaller angle is \(\min(\theta,360^\circ-\theta)\).
Standard events
In \(12\) hours, hands coincide \(11\) times, are opposite \(11\) times and are at right angles \(22\) times.
2.14 Age and Sequential Constraints
Age invariants
If present ages are \(A\) and \(B\), their difference stays constant:
\[(A+t)-(B+t)=A-B\]
Ratios change over time; differences do not.
Order chains
Translate \(A\) older than \(B\), \(B\) older than \(C\) as \(A\gt B\gt C\). Combine only compatible directions.
“Immediately” fixes a gap of one position; “before” does not.
2.15 Matrix and Missing-Number Puzzles
Row/column consistency
Test whether the same operation links cells across every row or every column. One row is not enough.
Common generators
Sum/difference, product, square, average, digit operations, alternating signs and positional weights.
Check symmetry and order.
Option verification
Insert each plausible option into the missing cell and demand that the rule work in all complete rows/columns.
2.16 Consistency and Uniqueness Audit
| Audit question | Why it matters |
|---|---|
| Does every entity appear exactly once where required? | Prevents duplication or omission |
| Does every direct clue hold? | Detects transcription errors |
| Does every negative clue hold? | Eliminates attractive but forbidden layouts |
| Are left/right and clockwise conventions consistent? | Prevents viewpoint reversal |
| Are all calculated regions/ages/distances valid? | Rejects negative, fractional or impossible results where inappropriate |
| Is the answer unique? | If not, a clue or case is unused |
3Short Tricks & Magic Formula Bank
3.1 Build Blocks
Immediate/adjacent clues become compact units; place the unit instead of separate entities.
3.2 Rank Formula
Total \(=L+R-1\). From one end to the other: \(R=N-L+1\).
3.3 Fix One on a Circle
Anchoring one person removes equivalent rotations and reduces cases sharply.
3.4 Generation Levels
Count vertical moves first, then determine gender and maternal/paternal branch.
3.5 Coordinate Route
Cancel opposite moves before applying Pythagoras to net \((\Delta x,\Delta y)\).
3.6 Modulo Week
Reduce any day interval modulo \(7\). Full weeks do not change the weekday.
3.7 Clock Angle
Use \(|30h-5.5m|\); if above \(180^\circ\), subtract from \(360^\circ\).
3.8 Age Difference
Age difference is invariant. Use it before forming ratio equations.
3.9 Compare Differences
In a number matrix, write row/column sums, gaps and products below it; patterns become visible.
3.10 Grid for Multi-Attribute
Rows are entities, columns are attributes; use ticks, crosses and one-to-one propagation.
3.11 Strongest Clue First
Absolute and immediate clues reduce more possibilities than vague negatives.
3.12 Reverse-Check
Read the completed model back into every clue; a single violation rejects it.
3.13 Formula Strip
| Problem | Formula / shortcut |
|---|---|
| Total from both-end rank | \(N=L+R-1\) |
| Between two same-end ranks | \(|r_1-r_2|-1\) |
| Positions with \(k\) between | \(|p_A-p_B|=k+1\) |
| Net route distance | \(\sqrt{(\Delta x)^2+(\Delta y)^2}\) |
| Weekday shift | \(N\bmod7\) |
| Clock angle | \(|30h-5.5m|\), then smaller angle |
| Age difference | \((A+t)-(B+t)=A-B\) |
| Circular arrangements (distinct, rotations same) | \((n-1)!\) |
4The SSC / TCS Traps — Red Flags
🚩 4.1 Left without Facing
Left/right reverses when viewpoint or facing direction changes.
🚩 4.2 Between vs Gap
One person between means a position difference of two, not one.
🚩 4.3 Inward/Outward Reversal
Clockwise meaning of left changes with facing.
🚩 4.4 Gender Inherited
A relation statement often reveals only one person's gender.
🚩 4.5 Final Direction vs Distance
Facing direction and direction of displacement are different outputs.
🚩 4.6 Century Leap Error
A century year must be divisible by \(400\), not merely \(4\).
🚩 4.7 Frozen Hour Hand
The hour hand moves \(0.5^\circ\) every minute; it is not fixed between hour marks.
🚩 4.8 Age Ratio Treated Constant
Age differences stay fixed; ratios usually change.
🚩 4.9 One-Row Pattern
A matrix rule must fit all corresponding rows/columns.
🚩 4.10 “Before” Becomes Immediate
Before/left of gives order only; “immediately” fixes adjacency.
🚩 4.11 Unused Clue
If a clue never influenced the model, the answer may not be unique.
🚩 4.12 Premature Branching
Use deterministic clue combinations before splitting into cases.
Puzzle guardrail: a neat layout is not enough. The final arrangement must satisfy every direct, negative, distance and exclusivity clue.
5Memory Hooks & Mnemonics
5.1 “SLOTS” Workflow
Setup, Link, Occupy, Test, Secure uniqueness.
5.2 “Immediate Makes a Brick”
Adjacent entities become a single movable block.
5.3 “Count the Person Once”
Both-end ranks add, then subtract one duplicated person.
5.4 “Pin the Circle”
Fix one person so the arrangement stops spinning.
5.5 “Family Is an Elevator”
Parents go up, children go down, spouse/sibling stays on the same floor.
5.6 “Route Bends, Displacement Cuts”
The shortest distance is the straight diagonal between start and finish.
5.7 “Weeks Vanish in Sevens”
Discard full weeks; only odd days shift the weekday.
5.8 “Thirty Hours Minus Five-and-Half Minutes”
The phrase recalls the clock-angle formula.
Arrangement Recall
- Number slots and mark facing.
- Place anchors and build blocks.
- Convert ranks correctly.
- Delay branching.
- Audit all clues.
Relationship Recall
- Family: generation before gender.
- Route: coordinates before distance.
- Calendar: reduce modulo seven.
- Clock: remember moving hour hand.
- Matrix: demand one global rule.
Final recall line: “Draw the model, lock the strongest clue, combine constraints, and verify the whole arrangement.”