DILR guide, section 9

Matching grid and truth-tellers

Both types are pure logic with no numbers. A matching grid falls to elimination, and a truth-teller set falls to turning each statement into a relation between people; set up correctly, each takes under 10 minutes.

Worked set 9, easy to medium. Attempt it in 15 minutes first.

Matching grid

Four friends, Anil, Bela, Chris and Dev, each live in a different city (Chennai, Delhi, Kolkata, Mumbai), do a different job (architect, banker, chef, doctor) and own a different pet (cat, dog, fish, parrot).

  1. 1The banker lives in Delhi.
  2. 2Bela owns the parrot.
  3. 3Whoever lives in Mumbai owns the fish.
  4. 4Anil is the chef and does not live in Chennai.
  5. 5Dev lives in Kolkata.
  6. 6The doctor owns the dog.
  7. 7The architect owns the cat.
  8. 8Dev does not own the dog.

Questions

  1. Who lives in Chennai?
  2. Which pet does the banker own?
  3. Which of these is true? (a) Dev is the doctor (b) The architect lives in Kolkata (c) Anil lives in Delhi (d) Chris owns the cat
  4. (TITA) If clue 8 were removed, how many complete assignments would satisfy the other clues?
Show solution
  • Dev lives in Kolkata (clue 5). Anil is the chef, so he is not the banker and cannot live in Delhi (clue 1); he is not in Chennai either (clue 4). So Anil lives in Mumbai and owns the fish (clue 3).
  • Bela owns the parrot, so she is neither the doctor (dog owner) nor the architect (cat owner). She is the banker, in Delhi, which leaves Chennai for Chris.
  • Chris and Dev share the doctor and architect jobs and the dog and cat. Clue 8 gives Dev the cat, so Dev is the architect and Chris the doctor with the dog.
PersonCityJobPet
AnilMumbaiChefFish
BelaDelhiBankerParrot
ChrisChennaiDoctorDog
DevKolkataArchitectCat

Answers. (1) Chris. (2) A parrot. (3) (b). (4) 2: without clue 8, Chris and Dev could swap jobs and pets.

Takeaway. A clue that links two attributes (“the doctor owns the dog”) rules out whole rows at once: Bela’s parrot removed two jobs in a single step.

8 minutes, timed

Practice set 9: matching grid

Three colleagues, Ira, Jai and Kavi, each drive a different-coloured car (red, blue, white), work in a different department (sales, HR, IT) and arrive at a different time (8, 9 or 10 a.m.).

  1. 1The person in IT arrives at 8.
  2. 2Jai drives the blue car and is not in sales.
  3. 3The red car’s owner arrives at 10.
  4. 4Kavi arrives before Ira.
  5. 5Ira is in sales.
  6. 6The person in HR drives the white car.

Questions

  1. Who arrives at 9?
  2. What colour is the IT person’s car?
  3. If clue 6 were removed, who could arrive at 8?
Show answers

(1) Kavi. (2) Blue. (3) Jai or Kavi. Ira is in sales, so she cannot arrive at 8 (that is the IT slot), and Kavi arrives before her. Jai’s blue car rules him out of 10, and Kavi cannot be last because he arrives before Ira, so the red car at 10 belongs to Ira. Jai is in HR or IT, and clue 6 rules out HR, so Jai is IT at 8 and Kavi is HR at 9 with the white car. Without clue 6, Kavi could take IT at 8 with Jai in HR at 9.

Worked set 10, medium. Attempt it in 15 minutes first.

Truth-tellers and liars

Five people, A, B, C, D and E, are each either a knight, who always tells the truth, or a knave, who always lies.

  1. 1A says: “B is a knave.”
  2. 2B says: “C and D are both knights.”
  3. 3C says: “Exactly one of A and E is a knave.”
  4. 4D says: “B is a knight.”
  5. 5E says: “A and D are of the same type.”

Questions

  1. (TITA) How many knights are there?
  2. Which pair are both knights? (a) A and B (b) A and C (c) C and D (d) D and E
  3. A sixth person, F, says “A is a knave.” What is F?
  4. Using only A’s and D’s statements, what can you conclude about E?
Show solution
  • A’s statement makes A and B opposite types. D’s statement makes D the same type as B. So A and D are always opposite, which makes E’s claim false: E is a knave.
  • With E a knave, C’s claim (“exactly one of A and E is a knave”) is true exactly when A is a knight, so C is the same type as A.
  • C matches A and D is opposite to A, so C and D can never both be knights. B’s claim is false and B is a knave. Then A is a knight, C a knight and D a knave.
  • Check every statement: A’s is true, B’s false, C’s true (A is a knight, E a knave), D’s false and E’s false.

Answers. (1) 2, namely A and C. (2) (b). (3) A knave, because A is a knight. (4) E is a knave, because A and D are always of opposite types.

Takeaway. Turn each statement into a relation (“A is opposite to B”, “D matches B”). Chains of relations usually settle most people before you test a single case.

8 minutes, timed

Practice set 10: truth-tellers

Four people, P, Q, R and S, are each a knight (always truthful) or a knave (always lying).

  1. 1P says: “Q is a knight.”
  2. 2Q says: “R is a knave.”
  3. 3R says: “P and Q are both knaves.”
  4. 4S says: “Exactly one of us four is a knight.”

Questions

  1. (TITA) How many knights are there?
  2. Is S a knight or a knave?
  3. Who are the knights?
Show answers

(1) 2. (2) A knave. (3) P and Q. P’s statement makes P and Q the same type, and Q’s makes R the opposite of Q, so either P and Q are knights with R a knave, or the reverse. In the reverse case R is the only knight among P, Q, R, and S breaks it: as a knight, S would be a second knight, making the claim false; as a knave, S would leave exactly one knight, making the claim true. So P and Q are knights, R is a knave, and S is a knave whose claim of “exactly one” is false.