Practice Set — Arrangements and a Points Table

Two sets, and neither one resolves to a single possibility. That is not an oversight — it is the point. The floors set has exactly two arrangements consistent with all five clues, and the points table is consistent with four different sets of match results. So some questions below have definite answers, and one of them honestly does not, which is how real sets behave. A practice diet where every set resolves uniquely trains a candidate to expect uniqueness and then to manufacture it at minute nine. Work each set to the point where you know what is fixed and what is still open, and answer only from the fixed part. Both sets were solved by exhaustive search before the questions were written, so every "cannot be determined" here is a computed fact rather than a hedge.

Before You Start

  • Draw the floors as six boxes and fill only certainties. "P on the topmost floor" is absolute and goes in first. "Two floors between Q and R" means their numbers differ by three, and it does not say which is higher — so it produces a pair of possibilities, not a placement.
  • For the league, check the total before anything else. Six matches are played and each awards 2 points in all, so the column must add to 12. Here 5 + 4 + 2 + 1 = 12, so the table is internally consistent — and if it had not added up, the right answer to every question in the set would have been that the data is impossible.
  • An odd points total means an odd number of draws for that team. A win is 2 and a loss is 0, so only draws can make a team's total odd. W on 5 and Z on 1 have each drawn an odd number of matches, and with three matches each that is a strong constraint — it is where this set opens up.

Key takeaways

  • "Exactly two floors between X and Y" means their floor numbers differ by three, and it never says which of the two is higher.
  • Work a set until you know what is fixed and what is still open, then answer only from the fixed part. Inventing a unique solution is how a set is lost.
  • In a round robin of n teams, the points column must add to 2 × n(n − 1)/2. Check that sum before solving anything.
  • A team with an odd points total has drawn an odd number of matches, because wins and losses contribute only even numbers.
  • "Cannot be determined" is a real answer in DILR and is offered honestly. Verify it by finding two consistent cases that disagree, not by giving up.

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.

  1. Set 1. Six people — P, Q, R, S, T and U — live on six floors of a building, numbered 1 (lowest) to 6 (highest), one person per floor.

    1. P lives on the topmost floor.
    2. There are exactly two floors between Q and R.
    3. S lives on some floor below Q.
    4. T lives immediately above U.
    5. R does not live on the lowest floor.

    Who lives on floor 3?

    1. T
    2. U
    3. S
    4. Cannot be determined
    Show answer

    Answer: B — U

    P takes floor 6. Q and R differ by 3, so from floors 1–5 the possible pairs are (1,4), (2,5) and (4,1), (5,2) — and R cannot be on floor 1, which removes two of them. S must be below Q, and T sits immediately above U. Working through, only two arrangements survive: S Q U T R P and S R U T Q P, reading floor 1 upward. U is on floor 3 in both, so the answer is definite even though the arrangement is not. This is the shape to look for — a question can be answerable in an unresolved set.

  2. Set 1. Six people — P, Q, R, S, T and U — live on six floors of a building, numbered 1 (lowest) to 6 (highest), one person per floor.

    1. P lives on the topmost floor.
    2. There are exactly two floors between Q and R.
    3. S lives on some floor below Q.
    4. T lives immediately above U.
    5. R does not live on the lowest floor.

    How many distinct arrangements satisfy all five clues?

    1. Exactly one
    2. Exactly two
    3. Exactly three
    4. More than three
    Show answer

    Answer: B — Exactly two

    Two: S Q U T R P and S R U T Q P from floor 1 upward. The clues fix S on floor 1, U on 3, T on 4 and P on 6 in both cases; what stays open is only whether Q is on 2 and R on 5 or the other way round, and no clue distinguishes those — S below Q holds either way, and R avoids floor 1 in both. A set with two solutions is common in CAT, and knowing there are two is what stops you answering the questions that depend on the difference.

  3. Set 1. Six people — P, Q, R, S, T and U — live on six floors of a building, numbered 1 (lowest) to 6 (highest), one person per floor.

    1. P lives on the topmost floor.
    2. There are exactly two floors between Q and R.
    3. S lives on some floor below Q.
    4. T lives immediately above U.
    5. R does not live on the lowest floor.

    If Q lives on floor 5, who lives on floor 2?

    1. R
    2. S
    3. T
    4. U
    Show answer

    Answer: A — R

    The extra condition picks out the second of the two arrangements, S R U T Q P, so floor 2 is R. This is why a conditional question ("if …") appears in almost every set that has multiple solutions: the condition resolves the branch, and a candidate who has already found both arrangements answers it in five seconds. A candidate who forced a single solution earlier will answer whichever one they invented.

  4. Set 1. Six people — P, Q, R, S, T and U — live on six floors of a building, numbered 1 (lowest) to 6 (highest), one person per floor.

    1. P lives on the topmost floor.
    2. There are exactly two floors between Q and R.
    3. S lives on some floor below Q.
    4. T lives immediately above U.
    5. R does not live on the lowest floor.

    Which of the following must be true?

    1. R lives above Q
    2. S lives on the lowest floor
    3. Q lives immediately below U
    4. T lives on floor 5
    Show answer

    Answer: B — S lives on the lowest floor

    S is on floor 1 in both arrangements, so that must be true. Option A holds in S Q U T R P and fails in S R U T Q P, which makes it possible rather than necessary — the distinction a "must be true" question exists to test. Option C holds only in the first arrangement too. Option D is false in both: T is on floor 4, immediately above U on 3.

  5. Set 2. Four teams — W, X, Y and Z — play a round robin in which every team plays every other team exactly once. A win gives 2 points, a draw 1 point and a loss 0. The final points are shown below.

    Final points table
    TeamPoints
    W5
    X4
    Y2
    Z1

    How many matches did W win?

    1. One
    2. Two
    3. Three
    4. Cannot be determined
    Show answer

    Answer: B — Two

    W has 5 points from three matches. Three wins would be 6, so W did not win all three; 5 must be 2 + 2 + 1, which is two wins and one draw. No other combination of 2s, 1s and 0s makes 5 from three matches — 2+1+1 is 4 and 1+1+1 is 3. The odd total is what forces the draw: only draws can make a total odd.

  6. Set 2. Four teams — W, X, Y and Z — play a round robin in which every team plays every other team exactly once. A win gives 2 points, a draw 1 point and a loss 0. The final points are shown below.

    Final points table
    TeamPoints
    W5
    X4
    Y2
    Z1

    Which team certainly won no match?

    1. Y
    2. Z
    3. Both Y and Z
    4. Cannot be determined
    Show answer

    Answer: B — Z

    Z has 1 point, which cannot include a win — a win alone is worth 2. So Z's single point is one draw and two losses, and Z won nothing. Y, on 2 points, is different: 2 can be one win and two losses or two draws and one loss, and both occur among the consistent result sets, so Y cannot be pinned down. Option C over-claims by treating Y like Z, which is exactly the reasoning to resist.

  7. Set 2. Four teams — W, X, Y and Z — play a round robin in which every team plays every other team exactly once. A win gives 2 points, a draw 1 point and a loss 0. The final points are shown below.

    Final points table
    TeamPoints
    W5
    X4
    Y2
    Z1

    How many of the six matches ended in a draw?

    1. Exactly one
    2. Exactly two
    3. Exactly three
    4. Cannot be determined
    Show answer

    Answer: D — Cannot be determined

    The points table is consistent with four different sets of results, and the number of drawn matches is 1 in one of them, 2 in two of them and 3 in the fourth — so it genuinely cannot be determined. What is fixed is that W drew exactly once and Z drew exactly once, from their odd totals. The lesson is that "how many draws in total" and "did this team draw" are different questions with different answerability, and a set will often settle the second while leaving the first open.

  8. Set 2. Four teams — W, X, Y and Z — play a round robin in which every team plays every other team exactly once. A win gives 2 points, a draw 1 point and a loss 0. The final points are shown below.

    Final points table
    TeamPoints
    W5
    X4
    Y2
    Z1

    Before solving anything, what does the points column tell you about the table's consistency?

    1. It is consistent: the points add to 12, which is 2 points × 6 matches
    2. It is inconsistent: the points should add to 24
    3. It is inconsistent: no team can have an odd number of points
    4. Nothing — the total is not constrained
    Show answer

    Answer: A — It is consistent: the points add to 12, which is 2 points × 6 matches

    Four teams play C(4,2) = 6 matches, and every match distributes exactly 2 points whether it is won or drawn. So the column must total 12, and 5 + 4 + 2 + 1 = 12 ✓. That check takes five seconds and is worth doing first: an inconsistent table means the whole set is broken, and it does happen in badly written practice material. Option C is a real misconception — odd totals are perfectly possible and simply mean an odd number of draws.