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.

• Game with Piles of Stones

  Two players are playing the following game. On the table are three piles of stones. The first pile has `10` stones, the second – `15`, and the third – `20`. Each player, in their turn, chooses one of the piles currently on the table and divides it into two smaller piles. The player who cannot make a move loses.

  Which of the two players has a winning strategy, and what is it?

  Topics:

  Combinatorics -> Invariants Combinatorics -> Game Theory Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Question

  On an island live liars and truth-tellers (truth-tellers always tell the truth, and liars always lie). What question should you ask a random person from the island's inhabitants to find out if they keep a crocodile as a pet?

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  Topics:

  Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems
  Sources:
  • problems.ru web site

• Question

  A knight exited the square `a1` and, after several moves, returned to the same square.

  Is it possible that the knight made an odd number of moves?

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  Topics:

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

• Question

  A knight moves from square `a1` to square `h8`. Is it possible that along the way it visited every square on the board exactly once?

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  Topics:

  Combinatorics -> Double Counting Combinatorics -> Invariants Combinatorics -> Graph Theory Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings -> Chessboard Coloring

• Question

  Given an `M times N` matrix, where each cell contains a real number. It is known that the sum of the numbers in each row and each column is equal to `1`.

  Prove that `M = N`.

  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Algebra -> Inequalities -> Averages / Means

• Question

  The numbers `1`, `2`, `3`, ..., `8` are written on the vertices of a cube. Prove that there exists an edge of the cube such that the difference between the numbers at its endpoints is at least `3`.

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Combinatorics -> Pigeonhole Principle Combinatorics -> Graph Theory Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction

• Question

  The game takes place on an infinite plane. One player moves the wolf, and another player moves K sheep. After the wolf's move, one of the sheep makes a move, then the wolf again, and so on. In one move, the wolf or a sheep cannot move more than one meter in any direction. Can the wolf always catch at least one sheep, regardless of the initial positions?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Geometry -> Plane Geometry Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 2

• Question

  K friends simultaneously learn K pieces of news (one piece of news per friend). They begin to phone each other and exchange news. Each call lasts one hour. How long will it take for all friends to know all the news? Consider the cases:
  (a) (5 points) K=64
  (b) (10 points) K=55
  (c) (12 points) K=100
  (a) Answer

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 4

• A mistake in the exercise

  Prove that there is an error in the following multiplication problem:

  \(\begin{array}& & & * & * & * & 2 & 7 \\ \times & & & & & * & * \\ \hline & * & * & * & * & * & 6 \\ + & * & * & * & * & * & \\ \hline & * & * & * & * & 4 & 6 \end{array}\)

  Topics:

  Arithmetic Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 4

• Pine Trees in the Forest

  A forester counts pine trees in a forest. He walked along each of the circles in the image, and within each circle he counted exactly `3` pine trees. Prove that the forester surely made a mistake in his count.

  

  Topics:

  Logic -> Reasoning / Logic Set Theory Proof and Example -> Proof by Contradiction