DILR guide, section 6

Scheduling and tournament

Both sets open with a counting equation that removes most of the guesswork: total hours in the schedule, total points in the tournament. Write that equation before placing anything.

Worked set 3, easy to medium. Attempt it in 15 minutes first.

Scheduling with durations

Arun, Bina, Chetan and Dia share one washing machine on a Sunday from 8 a.m. to 2 p.m. Each uses it once, for a whole number of hours, back to back with no gaps, and together they fill all 6 hours.

  1. 1Bina’s wash takes twice as long as Arun’s.
  2. 2Chetan starts at 10 a.m.
  3. 3Dia uses the machine for exactly 1 hour and is not the first user.
  4. 4Arun finishes before Chetan starts.

Questions

  1. Who uses the machine last?
  2. (TITA) At what hour does Dia finish?
  3. (TITA) If clue 4 were removed, how many schedules would be possible?
  4. Without clue 4, which of these is certain? (a) Arun goes first (b) Bina goes last (c) Dia finishes at 10 a.m. (d) Chetan uses the machine for 2 hours
Show solution

Durations: Arun a, Bina 2a, Dia 1, Chetan c. So a + 2a + 1 + c = 6, or 3a + c = 5, which forces a = 1 and c = 2.

  • Chetan runs 10–12, leaving a 2-hour block before (8–10) and one after (12–2).
  • The 8–10 block holds either Bina alone or Arun and Dia together.
  • Clue 4 puts Arun before 10, so 8–10 is Arun and Dia. Dia is not first, so Arun runs 8–9 and Dia 9–10. Bina takes 12–2.
Time8–99–1010–1212–2
UserArunDiaChetanBina

Without clue 4, Bina could also take 8–10, with Arun and Dia filling 12–2 in either order. That gives 3 schedules in all.

Answers. (1) Bina. (2) 10 (a.m.). (3) 3. (4) (d): the durations are fixed whatever the order.

Takeaway. One equation from the durations usually leaves only a handful of possibilities. Then place the fixed time (Chetan) and fill the blocks around it.

12 minutes, timed

Practice set 3: scheduling

Five talks, on AI, Banking, Climate, Design and Economics, fill five consecutive one-hour slots, numbered 1 to 5, at a conference.

  1. 1The Climate talk is immediately before the Design talk.
  2. 2Banking is in slot 2 or slot 4.
  3. 3AI is in neither the first nor the last slot.
  4. 4Economics comes after Banking.

Questions

  1. Which talk is in the middle slot?
  2. Which talk is last?
  3. (TITA) In which slot is the Design talk?
  4. (TITA) If clue 3 were removed, how many schedules would be possible?
Show answers

(1) AI. (2) Economics. (3) 2. (4) 4. Banking in slot 2 fails: Climate–Design would take 3–4 or 4–5, which either leaves AI only an end slot or pushes Economics into slot 1. So Banking is 4 and Economics 5; Climate–Design must be 1–2 (2–3 would put AI in slot 1), and AI takes 3. Without clue 3, these four work: AI–Banking–Climate–Design–Economics, AI–Banking–Economics–Climate–Design, Climate–Design–AI–Banking–Economics and AI–Climate–Design–Banking–Economics.

Worked set 4, medium. Attempt it in 15 minutes first.

Round-robin tournament

Four teams, P, Q, R and S, play a round robin: every team plays every other team once. A win earns 3 points, a draw 1 and a loss 0. The final points are P 7, Q 5, R 4 and S 0.

Questions

  1. (TITA) How many matches ended in a draw?
  2. Which team did P fail to beat?
  3. Which team or teams did R beat?
  4. If a win had earned 2 points instead of 3, with the same results, what would each team’s points be?
Show solution

Four teams play 4 × 3/2 = 6 matches. With w decisive matches and d draws: w + d = 6 and 3w + 2d = 7 + 5 + 4 + 0 = 16. So w = 4 and d = 2.

  • S has 0 points, so S lost all three of its matches.
  • Q earned 3 by beating S. Its other 2 points cannot include a win (that would add 3), so Q drew with both P and R.
  • P earned 3 (S) + 1 (draw with Q) = 4, so it beat R to reach 7.
  • R: 3 (S) + 1 (draw with Q) + 0 (lost to P) = 4, which checks out.
Row team vs column teamPQRS
P—DrawWonWon
QDraw—DrawWon
RLostDraw—Won
SLostLostLost—

Answers. (1) 2. (2) Q (a draw). (3) S only. (4) P 5, Q 4, R 3, S 0, so the order stays the same.

Takeaway. The total-points equation tells you how many draws there were before you place a single result. Then start from the extreme team (S with 0 points) and the team whose points allow only one split (Q).

15 minutes, timed

Practice set 4: tournament

Five teams, A, B, C, D and E, play a round robin. A win earns 3 points, a draw 1 and a loss 0. The final points are A 10, B 7, C 6, D 4 and E 1. E’s only point came from its match against B.

Questions

  1. (TITA) How many matches were drawn?
  2. Which team did D beat?
  3. Which of these is true? (a) C beat B (b) A drew with D (c) B lost to C (d) E drew with A
  4. (TITA) How many matches did A and B lose between them?
Show answers

(1) 2. (2) E. (3) (b). (4) 1. There are 10 matches, so w + d = 10 and 3w + 2d = 28, giving 8 wins and 2 draws, which means 4 “draw slots” across teams. A’s 10 points can only be 3 wins and a draw; B’s 7 only 2 wins, a draw and a loss; E’s 1 is a single draw. That is three slots; the fourth must be D’s (1 win, 1 draw, 2 losses), because C’s options give 0 or 3 draws. E drew with B, so the other draw is A–D. Then A beat B, C and E; B beat C and D; C beat D and E; D beat E.