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.

• The broken chain

  There are five segments of a broken chain, each segment having three links. Moses wants to repair the chain. What is the minimum number of links he needs to open and close again to join all these segments together?

  Note: The chain is not circular!

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems

• Question

  There are two coins in a wallet, and their sum is `15` shekels. One of the coins is not a `5` shekel coin. What are the coins in the wallet?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic

• Question

  a. The apple is heavier than the peach, and the peach is heavier than the pear. Which is heavier – the apple or the pear?

  b. The orange is heavier than the pear, and the plum is lighter than the orange. Which is heavier – the pear or the plum?

  Topics:

  Arithmetic Algebra -> Inequalities Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Question

  There are `85` balloons in a room, red and blue. It is known that:

  - At least one of the balloons is red,

  - In any pair of balloons we take, at least one of the balloons must be blue.

  How many red balloons are in the room?

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  One-seventh of mathematicians are philosophers, and one-ninth of philosophers are mathematicians. Which are there more of – philosophers or mathematicians?

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  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Fractions
  Sources:
  • problems.ru web site

• Question

  Is it possible to arrange all the numbers from `1` to `100` in a row such that the difference between any two adjacent numbers is at least `50`? If so, provide an example; if not, prove why.

  Topics:

  Number Theory Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  Is it possible to tile a `5xx5` board with dominoes?

  Note: The size of a board square matches the size of a domino square.

  Topics:

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

• Question

  64 friends were told 64 news items at the same time (one news item per friend). They start calling each other and exchanging news. Each call lasts one hour. How long will it take for all the friends to know all the news? 
  א. אנג'אנס 

  Topics:

  Logic -> Reasoning / Logic
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 4 Points 8

• Continue the Sequence

  `2, 6, 12, 20, 30…`

  a. What is the next number (the sixth) in the sequence?

  b. What is the `2016`th number in the sequence?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence
  Sources:
  • problems.ru web site

• Question

  Can you fill a `5xx5` table with real numbers such that the sum of each row is positive, and the sum of each column is negative?

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  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site