Logic, Reasoning / Logic

This category emphasizes general logical reasoning skills, often applied to puzzles or scenarios not strictly formal. It involves deduction, inference, identifying patterns, and drawing sound conclusions from given information. It overlaps with formal logic but can be broader.

• Baobab

  In the following exercise, identical digits have been replaced with identical letters, and different digits have been replaced with different letters. Reconstruct the exercise.

  `BAOxxBAxxB = 2002`

   

   

   

   

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Question

  In a certain country, there are more than 101 cities. The capital is connected by flight routes to 100 cities, and every city other than the capital is connected by flight routes to exactly 10 cities. It is given that from any city, it is possible to reach any other city (possibly not by a direct route). Prove that it is possible to close half of the flight routes leading to the capital such that the possibility of reaching any city from any other city is preserved.

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  Topics:

  Combinatorics -> Graph Theory Logic -> Reasoning / Logic Proof and Example
  Sources:
  • Tournament of Towns, 1981-1982, Spring

• Cats for Grandma

  Grandma Hannah has a number of cats. One day, her three grandchildren, Avi, Benny, and Gili, came to visit and tried to guess how many cats she has.

  Avi said: "Grandma has at least 7 cats",

  Benny said: "But less than 10, I think",

  Then Gili said: "Grandma Hannah has either 9, or 10, or 11 cats".

  "You are all correct!" replied the grandma

   So how many cats does she have?
   

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 4

• What is in each container?

  On the table, a cup, glass, pitcher, and jar are arranged in a row in an unknown order. The containers hold milk, orange juice, cola, and water, but it is unknown which liquid is in each container. Given that:

  • The milk and water are not in the cup.
  • The container holding orange juice is between the pitcher and the container holding cola.
  • The jar does not contain water or orange juice.
  • The glass is between the jar and the container with the milk.

  Which liquid is in which container?

   

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  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables

• Question

  Grandma Hannah has many flowerpots with flowers in her house. One day, her three grandchildren, Avi, Beni, and Gili, came to visit and tried to guess how many flowerpots she has.
  Avi said: "Grandma has more than 8 flowerpots,"
  Beni said: "More than 10, I think,"
  And then Gili said: "Grandma Hannah has at least 12 flowerpots."
  "Two of you are right, and one of you is wrong," answered the grandmother. So how many flowerpots does she have in the house?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 3

• Toys

  Jonathan has a collection of wooden toys. Some are cubes and some are spheres, some are red and some are blue.

  It is known that there are more spheres than cubes, and it is known that there are more blue toys than red toys.

  Prove that Jonathan has a blue sphere.

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 1

• The Magic Octopuses

  In the magic sea live octopuses who can talk. Each octopus either always tells the truth or always lies. One day
  the following conversation took place between four octopuses, Avi, Benny, Gidi, and Danny:

  
  Avi: I am a green octopus
  Benny: I am not green
  Gidi: All green octopuses are liars
  Danny: Only a green octopus can be a liar

  
  It is known that only one of these four is a liar, and the rest are truthful.
  a. Who is the liar among the four friends? Explain!
  b. Is it possible to know what his color is?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 4

• Log of Wood

  You have a very long log of wood. Can you measure exactly one meter from it, if you have for this purpose:
  а. A stick with a length of one and a half meters and another stick with a length of 40 centimeters,
  б. A stick with a length of one and a half meters and another stick with a length of 30 centimeters,

  Assuming you have no other measuring tools? Explain!

  Topics:

  Combinatorics -> Invariants Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 7
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 2

• Quiz

  In a class of 25 students, a quiz was given consisting of 7 questions. Prove that at least one of the following two statements is true:

  1. There is a student who solved an odd number of questions.
  2. There is a question that was solved by an even number of students.
  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 5

• Circle of Liars - The Truth Claim

  In a circle, `n` people are seated, each of whom is either a liar or a truth-teller.

  The people are looking towards the center of the circle. A liar always lies, and a truth-teller always tells the truth.

  Each of the people knows exactly who is a liar and who is a truth-teller.

  Each of the people says that the person sitting two places to their left (that is, next to the person sitting next to them), is a truth-teller.

  It is known that in the circle there is at least one liar, and at least one truth-teller.

  a. Is it possible that `n = 2017`?

  b. Is it possible that `n = 5778`?

  (Solution format: "word, word" for example "cat, puppy")

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  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems
  Sources:
  • Gillis Mathematical Olympiad, 2017-2018 Question 1