Logic

Logic is the study of reasoning and valid inference. It involves analyzing statements, arguments, and deductive processes. Questions may include solving logic puzzles, evaluating the truth of compound statements, using truth tables, and identifying logical fallacies.

• Question

  In every forest with bears, there are no wolves, and in every forest with wolves, there are also foxes. In the magical forest, there are foxes. Is it possible that there are also bears there?

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  Topics:

  Logic -> Reasoning / Logic

• Question

  Along the street are located `6` trees. One day, `6` parrots arrived and sat on the trees, one parrot on each tree. From time to time, two parrots each move to a neighboring tree of their choice. Can the parrots all gather on the same tree?

  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)

• Question

  In the following arithmetic puzzle, different digits have been replaced by different letters, and identical digits – by identical letters. Reconstruct the puzzle:

  `BAOxxBAxxB=2002`

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Inequalities Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Question

  Given `50` distinct natural numbers between `1` and `100`. It is known that no two of these numbers sum to `100`. Is it necessarily true that one of these numbers must be a perfect square?

  Topics:

  Number Theory -> Prime Numbers Arithmetic Combinatorics -> Pigeonhole Principle Combinatorics -> Matchings Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  Shlomi has a chessboard and a cube whose face size is the same as the size of a square on the board. Shlomi wants to paint the faces of the cube black and white, and then roll the cube across the board so that each time the face touching the board is the same color as the square it touches. The cube is supposed to pass through each square on the board exactly once. Can Shlomi do this? Justify or provide an example.

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Graph Theory Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Combinatorics -> Colorings -> Chessboard Coloring

• 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