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

  A `29×29` table contains all integers from `1` to `29`, each appearing exactly `29` times. The sum of all numbers above the main diagonal is exactly three times greater than the sum of all numbers below the main diagonal. What number is written in the central cell of the table?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Combinatorics -> Number Tables Minimum and Maximum Problems / Optimization Problems

• Question

  From a chessboard, two opposite corners are removed (the squares `a1` and `h8`, for example). Can you tile the remaining board with dominoes?

  Show hint

  Log in here to receive the hint
  Topics:

  Combinatorics -> Invariants Combinatorics -> Matchings Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings -> Chessboard Coloring Puzzles and Rebuses

• Ants on a Stick

  On a stick that is one meter long, there are `10` ants, `5` from each side, at distances of one centimeter (see the picture). The ants on the left side of the stick move to the right, and the ants on the right side of the stick move to the left. The speed of each ant is constant and equal to one centimeter per second. When two ants meet, they both reverse direction and start moving away from each other. When any ant reaches the end of the stick, it falls off (ants are particularly stupid creatures).

  

  a. Will all the ants fall off the stick, and if so, in how much time?

  b. How many collisions will occur between the ants?

  Topics:

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

• Bacteria in a Test Tube

  A scientist has a test tube containing bacteria. Every second, each bacterium splits into two. After two hours, the test tube was completely full of bacteria. How long before that was the test tube exactly half full?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards

• Question

  Find the sum of all natural numbers from `1` to `100`.

  Topics:

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

• Camels and Horses

  In a herd, there are dromedary camels (one-humped), Bactrian camels (two-humped), and horses. It is known that the total number of humps in this herd is `200`. In addition, the number of horses is equal to the number of Bactrian camels. How many animals are there in the herd in total?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic

• Milk and Coffee

  A. Hannah has two cups, one with milk and the other with coffee. The amounts of milk and coffee are the same. Hannah transfers two teaspoons of milk into the cup with coffee, mixes, and then transfers two teaspoons from the cup of coffee into the milk. Currently, she has two cups each containing a mixture of coffee and milk. What is greater – the amount of milk in the cup of coffee or the amount of coffee in the cup of milk?

  B. The same question, but this time Hannah adds milk to the coffee without mixing.

  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic

• Question

  Hannah is waiting for a bus. Which of the following three events is most likely to occur:

  1. Hannah waits for the bus for at least one minute,
  2. Hannah waits for the bus for at least two minutes,
  3. Hannah waits for the bus for at least five minutes.
  Topics:

  Logic -> Reasoning / Logic Set Theory Probability Theory

• Watering Potted Plants

  Hannah and Rachel watered potted plants at home. It is known that each of them watered exactly half the number of potted plants in the house, and it is also known that three potted plants were accidentally watered twice. How many potted plants remain to be watered?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Set Theory

• Knights of the Round Table

  Around a round table sit 12 knights, each of whom is either an elf or a dwarf. It is known that the number of elves is greater than the number of dwarves. Prove that there are two elves who sit opposite each other.

  Will this continue to be true if the total number of knights is 120?

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction