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.

• 9 Coins

  Given `9` coins that look identical. One of the coins is counterfeit, and its weight is less than the weight of a regular coin. How to find the counterfeit coin using two weighings on a balance scale without weights?

  Topics:

  Logic -> Reasoning / Logic Algorithm Theory -> Weighing Minimum and Maximum Problems / Optimization Problems

• 26 Coins

  Given `26` coins that look identical. One of the coins is counterfeit, and it weighs less than a regular coin. How can you find the counterfeit coin using three weighings on a balance scale without weights?

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  Topics:

  Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Algorithm Theory -> Weighing Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems

• 80 Coins

  Given `80` coins that look identical. One of the coins is counterfeit and weighs less than a regular coin. How can you find the counterfeit coin using four weighings on a balance scale without weights?

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  Topics:

  Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Algorithm Theory -> Weighing Minimum and Maximum Problems / Optimization Problems

• Question

  By how much is the sum of all even numbers not exceeding `100` greater than the sum of all odd numbers not exceeding `100`?

  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Algebraic Techniques Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence

• Question

  A. You have a large jug of 12 liters of olive oil and two empty smaller vessels, one of 5 liters and one of 8 liters. Can you divide the oil you have into two equal parts, if you only have these vessels and no additional measuring tools?

  B. The same question, but instead of the 5-liter vessel, you have a 4-liter vessel.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  `120` identical spheres are arranged in the shape of a triangular pyramid. How many layers are there in the pyramid?

  Note: This is a pyramid, which is a three-dimensional shape, and not a triangle in a plane.

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  Topics:

  Geometry -> Solid Geometry / Geometry in Space Arithmetic Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence Number Theory -> Triangular Numbers

• Question

  A bouquet is composed of `7` roses, white and red (both colors are present). It is known that out of every two roses, one is necessarily white. How many white roses and how many red roses are in the bouquet?

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Each of seven children holds two balloons, one red and one yellow. Can they exchange balloons so that each child has two balloons of the same color?

  Topics:

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

• Question

  `19` apple trees are arranged in a circle. Prove that there exists a pair of adjacent trees such that the total number of apples on them is even.

  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  In the magical land, there are only coins of `4`, `7`, and `10` liras. Shlomi currently only has coins of `4` and `10` liras. Shlomi wants to buy a book that costs `23` liras. Will he be able to pay for the book without change?

  Topics:

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