Arrangements, Distributions, Games and Tournaments
This is the reasoning half of DILR, and it is worth learning as one method with several costumes rather than as a dozen puzzle types. Seating in a row, seating in a circle, people on floors of a building, boxes of different colours, matches in a tournament — all of them are the same three moves: work out what the grid should be, enter only what is certain, and when nothing more is certain, branch on the cell with the fewest options. A candidate who has learnt "types" is lost the moment a set does not match one. A candidate who has learnt the method starts every set the same way, which is the real advantage in a section where the sets are designed to be unfamiliar.
Rows, Circles and Floors
- Rows: fix the direction once, in writing. Draw the row and label the left end. "To the right of" means the reader's right unless the set says the people are facing you — in which case every left and right reverses. Write "facing north →" at the top of your diagram and every condition after that is unambiguous.
- Circles: only relative position matters, so fix one person and never move them. Then "third to the left of P" is countable. Whether the people face the centre or away from it flips every direction, so mark that on the diagram before entering a single condition — it is the most common single cause of a wholly wrong circular arrangement.
- Floors: number them and treat "above" as a larger number. "Exactly two floors between P and Q" means |P − Q| = 3, not 2 — the two floors are between them, so the gap in numbering is one more. That off-by-one is deliberately planted in almost every floor set.
- Worked fragment. Five people P, Q, R, S, T sit in a row facing north. R is third from the left. Q is immediately to the right of R. S is at one of the ends. P is not next to R. Start with the absolute: R is at position 3, so Q is at 4. S is at 1 or 5. If S is at 1, then P and T take 2 and 5; P cannot be at 2 (next to R at 3), so P is at 5 and T at 2 — a complete arrangement, S T R Q P. If S is at 5, then Q is at 4 and S at 5 leaves 1 and 2 for P and T; P cannot be at 2, so P is at 1 and T at 2, giving P T R Q S. Two valid arrangements, and a question asking "who is at the extreme left?" would be answerable only if a further condition rules one out — which is exactly the kind of thing to check before assuming a unique answer.
Distribution Grids
When the set matches people to two or more attributes — name, city, sport, salary — the shape is a table with one row per person and one column per attribute. Two rules make it mechanical.
- One tick per row and per column, when the matching is one-to-one. The moment a cell is ticked, every other cell in its row and its column is crossed. Doing that immediately, every time, is what turns four conditions into a solved grid — the eliminations do the work, not the ticks.
- Read "or" carefully. "P likes either cricket or hockey" means one of them, not both, and it eliminates every other sport for P in one stroke. A condition that looks vague is often the most powerful one in the set.
- Counting conditions are conditions. "Exactly two of the five are from Mumbai" constrains the whole column and is easy to forget once you start chasing individuals. Note such totals at the top of the column before you begin.
Games and Tournaments
- Round robin: n teams play n(n − 1)/2 matches, each team playing n − 1 games. With 5 teams that is 10 matches. The total number of wins across all teams equals the number of matches, which is the check that catches an inconsistent table.
- Knockout: to produce one champion from n teams, n − 1 teams must be eliminated, so there are exactly n − 1 matches. No case work is needed for that count, whatever the bracket looks like — each match eliminates exactly one team.
- Points tables: write the arithmetic of the points system down first. With 2 points for a win, 1 for a draw and 0 for a loss, a team on 7 points from 5 games has (3 wins, 1 draw, 1 loss) or (2 wins, 3 draws, 0 losses) — enumerate both rather than assuming. The number of draws is always even across the table, because each drawn match contributes a draw to two teams.
- "Best possible" and "worst possible" are the real question type. "What is the minimum number of points with which a team can still qualify?" is answered by constructing the most favourable arrangement of other teams' results — not by computing an average. Build the extreme case explicitly; it is always constructible, and the question is asking you to construct it.
⚠️ Two teams can share a rank
Unless the set gives a tie-breaker, equal points means an undetermined order — and a question asking "who finished second?" then has no unique answer. Check for the tie-breaking rule before answering any ranking question; its absence is sometimes the point of the set.Key takeaways
- Every set of this kind is the same three moves: choose the grid, enter only certainties, then branch on the tightest constraint.
- Start from a condition that names an actual position, not a relation. One absolute condition is worth three relative ones.
- Crosses solve sets, not ticks. Every negative condition belongs on the grid, because eliminations are what eventually force an answer.
- In a circle, fix one person and mark whether everyone faces inward or outward before entering any condition — that single note prevents a wholly reversed solution.
- "Exactly two floors between P and Q" means their numbers differ by three. Read every gap statement for that off-by-one.
- Round robin with n teams is n(n − 1)/2 matches; a knockout is n − 1 matches, because each match eliminates exactly one team.
- A "minimum points to qualify" question is asking you to construct the most favourable case, not to average anything.
- Two teams on equal points have no determined order unless a tie-breaker is given, so a ranking question may have no unique answer.
Practice questions (8)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
In a round-robin tournament where every team plays every other team exactly once, 8 teams take part. The number of matches is:
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Answer: C — 28
Each match is a pair of teams, so the count is C(8,2) = 8 × 7/2 = 28. Option B, 56, is 8 × 7 — it counts each match twice, once from each team's side. Option D, 7, is the number of games one team plays.In a knockout tournament with 37 teams (some receiving byes), the total number of matches needed to decide a single winner is:
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Answer: A — 36
Every match eliminates exactly one team, and 36 teams must be eliminated to leave one champion, so exactly 36 matches are played. The bracket structure and the byes change who plays whom, but not the count — which is why option D, though it sounds cautious, is wrong.Five friends live on five floors of a building, numbered 1 (lowest) to 5. There are exactly two floors between A and B. Which pair of floors could A and B occupy?
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Answer: B — 1 and 4
Two floors between them means two floors strictly in the middle, so the floor numbers differ by 3. Floors 1 and 4 have floors 2 and 3 between them ✓. Option A, floors 1 and 3, has only floor 2 between them. Option D, floors 1 and 5, has three floors between. This off-by-one is deliberately planted in almost every floor-based set.Five people sit in a row facing north. R is third from the left, Q is immediately right of R, S is at one end, and P is not adjacent to R. How many arrangements are possible?
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Answer: B — 2
R is at 3 and Q at 4. S is at 1 or 5. If S is at 1, positions 2 and 5 remain for P and T; P cannot be at 2 (adjacent to R), so P is at 5, T at 2: S T R Q P. If S is at 5, positions 1 and 2 remain; P cannot be at 2, so P is at 1, T at 2: P T R Q S. Both satisfy every condition, so the answer is 2 — and noticing that there are two, rather than assuming a unique arrangement, is what a following question about "who sits at the left end" would depend on.In a league where a win gives 2 points, a draw 1 and a loss 0, a team has 7 points from 5 matches. Which is NOT a possible record?
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Answer: C — 1 win, 5 draws, 0 losses
Check the points and the match count together. Option A: 6 + 1 = 7 points from 5 matches ✓. Option B: 4 + 3 = 7 points from 5 matches ✓. Option C: 2 + 5 = 7 points, but that is 6 matches, not 5 — the points work and the match count does not, which is exactly the kind of half-consistent option these sets offer. Always verify both totals.Across a completed league table, the total number of drawn results recorded by all teams together must be:
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Answer: A — Even
A drawn match gives a draw to both teams, so every draw contributes 2 to the column total — which must therefore be even. This is the fastest consistency check on a partially filled league table: an odd draw column means a data-entry error somewhere. Option C is wrong because the draw total is twice the number of drawn matches, not equal to the number of matches played.Eight people sit around a circular table facing the centre. "Third to the left of P" is counted:
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Answer: A — Clockwise from P, as seen from above
For a person facing the centre, their left hand points clockwise as the table is seen from above. That is why the facing direction must be noted before any condition is entered: if the same people faced outward, every left would become anticlockwise and the entire arrangement would come out mirrored. Option D is the dangerous one — direction is exactly what does matter in a circle, since only relative position is fixed.A grid matches 4 people to 4 distinct cities, one each. You establish that P is in Delhi. The immediate next step is:
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Answer: B — Cross out Delhi for the other three people and every other city for P
In a one-to-one matching, a single tick forces six crosses in a 4 × 4 grid — three in the row and three in the column. Entering them immediately is what makes the next condition powerful, because it often has only one cell left to occupy. Candidates who postpone the eliminations end up re-deriving them, or worse, assigning two people to one city.