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.

• Question

  Can you fill a table of size `5xx5` with

  a. Integers,

  b. Real numbers,

   such that the sum of each row is even, and the sum of each column is odd?

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  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  Given a three-digit prime number with all its digits distinct. It is known that its last digit is equal to the sum of the other two digits. Find all the possibilities for the last digit of this number.

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  In two classes with an equal number of students, a quiz was administered. After grading the quiz, the teacher claimed that the number of grades of `0 ` was `13` greater than the number of all other grades combined. Is it possible that he was mistaken?

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  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• The Knight and the Dragon

  A knight encountered a dragon with three heads on his way and they began to fight. Every time the knight chops off one of the dragon's heads, three new heads appear in its place. Is it possible that at the end of the battle, the dragon will have a thousand heads?

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  Topics:

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

• Question

  Is it possible to make change for a `25` lira note using `10` coins worth `1`, `3`, and `5` lira?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • problems.ru web site

• Question

  When there are no classes and the weather is good, Danny always goes to the park. Today the weather is good, and Danny is going to the park. Is it necessarily the case that there are no classes today?

  Topics:

  Logic -> Reasoning / Logic

• Ali Baba and the Forty Thieves

  Ali Baba wrote the number `17` on a piece of paper. The forty thieves pass the paper to each other, and each one either adds `1` to the existing number, or subtracts `1`, until each of them has done so once, and then they return the paper to Ali Baba.

  Is it possible that the number now written on the paper is `40`?

  Topics:

  Arithmetic Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Find all pairs of prime numbers whose difference is `17`.

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  Topics:

  Number Theory -> Prime Numbers Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  A group of children went to an amusement park. It is known that the number of girls wearing hats is equal to the number of boys not wearing hats.

  Who are there more of in this group: children not wearing hats or girls?

  Topics:

  Logic -> Reasoning / Logic Set Theory
  Sources:
  • problems.ru web site

• Question

  In the magical land, in every school where both physics and geography are studied, astronomy is necessarily also studied.

  It is known that at Alice's school, astronomy and physics are studied. Do they also study geography?

  Topics:

  Logic -> Reasoning / Logic