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

  `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)

• Question

  Seven wolves eat seven sheep in seven days. How long will it take `9` wolves to eat `9` sheep?

  Topics:

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

• An Ancient Russian Riddle

  Two woodcutters, Ivan and Prokhor, were working together in the forest. They settled down to eat lunch. Ivan had `4` pitas, and Prokhor had `8`. At that moment, a hunter passed by them and asked to join the meal, so they divided the `12` pitas equally among the three of them. After the meal, the hunter really wanted to thank the woodcutters and gave them `15` kopecks. After he left, Ivan and Prokhor began to discuss how to divide the money.

  Prokhor suggested simply dividing it equally. But Ivan argued that this would not be fair, because he had twice as few pitas, therefore, in his opinion, Prokhor should receive `10` kopecks and Ivan himself - `5`. How do you think the money should be divided fairly?

  Topics:

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

• Question

  When there is heat stress or cold stress, military training is canceled. One day, training was canceled, and it is known that this did not happen because of the cold. Does this mean that the training was canceled because of the heat?

  Topics:

  Logic -> Reasoning / Logic

• Question

  Avi is a good student, but his brother Benny is less so. A day is considered successful if Avi received `100` and Benny did not receive zero. Define an unsuccessful day.

  Topics:

  Logic -> Reasoning / Logic

• Question

  What is the maximum number of rooks that can be placed on an `8xx8` board so that they do not threaten each other?

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems