Combinatorics, Case Analysis / Checking Cases, Processes / Procedures

Case analysis is a problem-solving technique where a problem is divided into several distinct, exhaustive cases. Each case is then analyzed separately to arrive at a solution or proof. Questions suitable for this involve conditions that naturally split the problem into different scenarios.

• Question

  A box contains `14` black socks and `14` white socks. Apart from color, all socks in the box are identical. Danny wants to take a pair of socks from the box without looking inside. How many socks does he need to take out to:

  A. Certainly have any pair of socks,

  B. Certainly have a pair of black socks?

  Topics:

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

• Question

  On the circle, there are blue and red points. It is allowed to add a red point and change the colors of its neighboring points or remove a red point and change the colors of its neighboring points (it is not allowed to leave fewer than 2 points on the circle). Prove that it is impossible to move, using only these operations, from a circle with two red points to a circle with two blue points.
  K. Kaznvosky

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Algebra Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings -> Chessboard Coloring
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 1

• Question

  Let a₁, a₂, ..., a₁₀₁ be a permutation of 2, 3, 4, ..., 102. Find all permutations such that a_i is divisible by i for all i.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 3

• Question

  A tourist arrived at an island inhabited by truth-tellers and liars. Truth-tellers always tell the truth, and liars always lie. The tourist is traveling with a local guide, and they see a farmer working in the field. The tourist asked the guide to find out if this man is a truth-teller or a liar. The guide spoke with the farmer and then said to the tourist: "He said that he is a truth-teller."

  A. What can be deduced about the farmer?

  B. What can be deduced about the guide?

  Topics:

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

• Question

  On Alice's birthday, her friends are asked how old she is. The Mad Hatter says that Alice's age is greater than `11`, and the Cheshire Cat says that her age is greater than `10`. It is known that exactly one of them is wrong. How old is Alice now? Explain!

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  Topics:

  Number Theory Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  One day, Harry Potter found a strange notebook in which the following one hundred sentences were written:

  "In this notebook, there is exactly one sentence that is false."

  "In this notebook, there are exactly two sentences that are false."

  "In this notebook, there are exactly three sentences that are false."

  ...

  "In this notebook, there are exactly one hundred sentences that are false."

  Are there any true sentences in this notebook, and if so, how many? Justify your answer!

  Topics:

  Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems Logic -> Reasoning / Logic -> Paradoxes

• Question

  Three hedgehogs have three pieces of cheese weighing `5`, `8`, and `11` grams. A fox offers to help the hedgehogs divide the cheese equally. The fox can bite off one gram from each of two cheese pieces of its choice. Can the fox, using these actions, reach a state where it leaves the three hedgehogs with equal pieces of cheese?

  Topics:

  Arithmetic Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  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

  How can you divide `7` apples equally among `12` children, if you are not allowed to cut any apple into more than `5` pieces?

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  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • problems.ru web site

• 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