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.

• The Beaver and the Mole

  There is a plot of land in the shape of a square `4 times 4` divided into cells of `1 times 1`. The beaver wants to build a house on it that occupies 4 cells, which from a top-down view looks like this:

  

  The mole wants to disturb him. For this purpose, it can dig holes, each of which occupies one cell. It is impossible to build on the cells that have become holes. What is the smallest number of holes the mole needs to dig so that the beaver cannot build the house?

   

  Topics:

  Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 4
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 2

• Doughnuts

  Ayala, Benny, Gili, Danny, and Hadas received a package of doughnuts containing

  • 10 doughnuts with dulce de leche
  • 8 doughnuts with peanut butter
  • 9 doughnuts with chocolate
  • 11 with strawberry jam

  Each of them has their favorite type of doughnut.

  • Ayala ate 5 doughnuts of her favorite type
  • Benny ate 6 doughnuts of his favorite type
  • Gili ate 7 doughnuts of her favorite type
  • Danny ate 8 doughnuts of his favorite type
  • Hadas ate 9 doughnuts of her favorite type

  After that, they were left with 3 doughnuts of different types. What is each of their favorite type of doughnut?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 5

• 6 on the Board

  The number 6 is written on the board. At each step, you can add the digit 6 to the end of the number (so that it is the units digit,) or replace the number with the sum of its digits.
  Which numbers can be obtained in this way? Describe the entire set of numbers and explain why there are no more

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 5

• Another Donkey in the Middle

  Avi, Beni, and Gadi played "Donkey in the Middle" - at any given moment, someone is in the middle, trying to catch a ball that the other two are passing to each other.
  If he succeeds, one of the other two replaces him. After the game, Avi said that he was in the middle 8 times,
  Beni said he was in the middle 4 times, and Gadi forgot how many times he was in the middle.
  They also remember that Beni was the last one in the middle. Describe all the possibilities for the number of times Gadi was in the middle,

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 3

• Ten-Digit Number

  Yael writes ten-digit numbers in whose decimal representation each of the digits `0, 1, 2, 3, 4, 5, 6, 7, 8, 9` appears exactly once.
  In the numbers that Yael writes, the difference between any two adjacent digits is at least 2. What is the smallest number Yael can write?
   

  Topics:

  Number Theory Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 3
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 1

• The Round Table

  Around a round table are 12 chairs, with knights sitting on some of them. Arthur wants to join the meeting,
  and it turns out that no matter where he sits, someone is definitely sitting next to him.
  What is the smallest number of knights that can be around the table to ensure this is true? (not including Arthur)

  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 1

• Two Hashes

  What is the maximum number of "domino" shapes (rectangles `1 times 2` or `2 times 1`) that can be placed inside the orange shape,
  such that they do not overlap and do not extend beyond the boundaries of the shape?

  

  Topics:

  Combinatorics -> Invariants Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Minimum and Maximum Problems / Optimization Problems Combinatorics -> Colorings -> Chessboard Coloring Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 2
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 1

• How many triangles?

  How many triangles are there in the picture?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Angle Calculation Number Theory -> Division
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 7

• The Number

  Given a positive integer less than 2000.
  If it is not divisible by 43, then it is divisible by 41,
  If it is not divisible by 53, then it is divisible by 43,
  If it is not divisible by 41, then it is divisible by 53.
  Find the number.

  Topics:

  Number Theory -> Prime Numbers Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 4

• SLV LVS BLS

  In the following expression, different letters represent different digits, and identical letters represent identical digits:

  SLV = LVS + BLS

  Find the number SLV.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 3