Linear Arrangement made clear.
Give each seat a number and decide which direction everyone faces. Then translate left, right, adjacency and gaps into precise position constraints.
Key concepts in Linear Arrangement
Facing north
On a left-to-right page diagram, a north-facing person’s left is the page’s left.
Facing south
A south-facing person’s left is the page’s right. The diagram’s coordinates do not change; the person-relative direction does.
Immediately beside
The positions differ by exactly one.
People between
If k people sit between A and B, their seat numbers differ by k+1.
Same seats, different facing directions
| Page seat | 1 | 2 | 3 |
|---|---|---|---|
| Person | A | B | C |
| All face north | A is left of B | B | C is right of B |
| All face south | A is right of B | B | C is left of B |
A reliable solving method
- Draw one numbered slot for each seat.
- Write the facing direction above the row.
- Place endpoints and immediate-neighbour blocks first.
- Test every remaining arrangement against all clues before selecting a definite answer.
Step-by-step solved examples
A,B,C sit in a row facing north. A is immediately left of B; C is right of B. Order?
- Treat A–B as an adjacent block.
- C must occupy a position after B.
- Only A B C fits the three seats.
In five seats, A is at seat 2 and two people are between A and B. Find B.
- Seat difference is 2+1 = 3.
- Candidates for B are 2−3 or 2+3.
- Only seat 5 is within the row.
P,Q,R face south; page order is P Q R. Who is to Q’s left?
- South-facing reverses personal left and page-left.
- Q’s left points toward the page’s right.
- R is to Q’s left.
Linear Arrangement: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
North-facing A B C: who is immediately right of B?
Show answer and explanation — question 1
Answer C: C
For north-facing people, page-right is personal right.
South-facing A B C: who is immediately right of B?
Show answer and explanation — question 2
Answer A: A
For south-facing B, personal right points to page-left.
Three people between A and B means seat difference?
Show answer and explanation — question 3
Answer C: 4
Include one endpoint step: 3+1 = 4.
A in seat 1, B immediately right, all north-facing. B’s seat?
Show answer and explanation — question 4
Answer B: 2
Immediate right is the next slot, seat 2.
Four seats: A at 1, D at 4, B immediately left of C. Order?
Show answer and explanation — question 5
Answer B: A B C D
B and C fill seats 2 and 3 in that order.
A is somewhere left of B. Must they be adjacent?
Show answer and explanation — question 6
Answer B: No
“Left of” does not specify the number of intervening people.
Frequently asked questions
How should I treat “second to the left”?
Move two seats in that person’s left direction; exactly one seat lies between.
What if several orders work?
Report only conclusions common to all valid orders, or choose “cannot be determined” if the requested fact varies.
Draw one numbered slot for each seat. Test every remaining arrangement against all clues before selecting a definite answer.