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Open Problems in Discrete Mathematics

Description: Welcome to the quiz on Open Problems in Discrete Mathematics! This quiz will test your knowledge and understanding of some of the most challenging and unsolved problems in the field.
Number of Questions: 10
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Tags: discrete mathematics open problems combinatorics graph theory number theory
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What is the Erdős-Rényi model?

  1. A random graph model where each pair of vertices has a probability p of being connected by an edge.

  2. A random graph model where each vertex has a probability p of being connected to every other vertex.

  3. A random graph model where each edge has a probability p of being present in the graph.

  4. A random graph model where each subgraph has a probability p of being present in the graph.

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Correct Option: A
Explanation:

The Erdős-Rényi model is a random graph model where each pair of vertices has a probability p of being connected by an edge. This model is widely used in the study of random graphs and has applications in various fields, including computer science, physics, and biology.

What is the P versus NP problem?

  1. The problem of determining whether a given problem can be solved in polynomial time.

  2. The problem of determining whether a given problem can be solved in exponential time.

  3. The problem of determining whether a given problem can be solved in logarithmic time.

  4. The problem of determining whether a given problem can be solved in constant time.

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Correct Option: A
Explanation:

The P versus NP problem is one of the most important and challenging open problems in computer science. It asks whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. If the answer is yes, then P = NP, which would have profound implications for the efficiency of many algorithms.

What is the Collatz conjecture?

  1. A conjecture that states that every positive integer will eventually reach 1 under the Collatz sequence.

  2. A conjecture that states that every positive integer will eventually reach a repeating cycle under the Collatz sequence.

  3. A conjecture that states that every positive integer will eventually reach a prime number under the Collatz sequence.

  4. A conjecture that states that every positive integer will eventually reach a perfect number under the Collatz sequence.

Show answer
Correct Option: A
Explanation:

The Collatz conjecture is a famous unsolved problem in mathematics. It states that for any positive integer n, if n is even, divide it by 2, and if n is odd, multiply it by 3 and add 1, then the sequence of numbers obtained will eventually reach 1. Despite extensive research, the conjecture remains unproven.

What is the four color theorem?

  1. A theorem that states that any map can be colored with four colors in such a way that no two adjacent regions have the same color.

  2. A theorem that states that any map can be colored with five colors in such a way that no two adjacent regions have the same color.

  3. A theorem that states that any map can be colored with six colors in such a way that no two adjacent regions have the same color.

  4. A theorem that states that any map can be colored with seven colors in such a way that no two adjacent regions have the same color.

Show answer
Correct Option: A
Explanation:

The four color theorem is a famous theorem in graph theory that states that any map can be colored with four colors in such a way that no two adjacent regions have the same color. The theorem was first conjectured in 1852 and was finally proven in 1976 using a computer-assisted proof.

What is the Goldbach conjecture?

  1. A conjecture that states that every even integer greater than 2 can be expressed as the sum of two primes.

  2. A conjecture that states that every even integer greater than 2 can be expressed as the sum of three primes.

  3. A conjecture that states that every even integer greater than 2 can be expressed as the sum of four primes.

  4. A conjecture that states that every even integer greater than 2 can be expressed as the sum of five primes.

Show answer
Correct Option: A
Explanation:

The Goldbach conjecture is a famous unsolved problem in number theory. It states that every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been verified for all even integers up to 4³10^{18}, but a general proof remains elusive.

What is the Twin Prime Conjecture?

  1. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 2.

  2. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 3.

  3. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 4.

  4. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 5.

Show answer
Correct Option: A
Explanation:

The Twin Prime Conjecture is a famous unsolved problem in number theory. It states that there are infinitely many pairs of prime numbers that differ by 2. The conjecture has been verified for all prime numbers up to 10^{18}, but a general proof remains elusive.

What is the Happy Number Conjecture?

  1. A conjecture that states that every positive integer eventually reaches 1 under the Happy Number sequence.

  2. A conjecture that states that every positive integer eventually reaches a repeating cycle under the Happy Number sequence.

  3. A conjecture that states that every positive integer eventually reaches a prime number under the Happy Number sequence.

  4. A conjecture that states that every positive integer eventually reaches a perfect number under the Happy Number sequence.

Show answer
Correct Option: A
Explanation:

The Happy Number Conjecture is a famous unsolved problem in number theory. It states that every positive integer eventually reaches 1 under the Happy Number sequence. The Happy Number sequence is defined as follows: for a given positive integer n, if n is even, divide it by 2, and if n is odd, square each digit of n and add the results together. Repeat this process until n reaches 1. The conjecture states that every positive integer will eventually reach 1 under this process.

What is the Strong Perfect Number Conjecture?

  1. A conjecture that states that there exists a perfect number that is also prime.

  2. A conjecture that states that there exists a perfect number that is also odd.

  3. A conjecture that states that there exists a perfect number that is also a square.

  4. A conjecture that states that there exists a perfect number that is also a cube.

Show answer
Correct Option: A
Explanation:

The Strong Perfect Number Conjecture is a famous unsolved problem in number theory. It states that there exists a perfect number that is also prime. A perfect number is a positive integer that is equal to the sum of its proper divisors (the divisors of the number excluding the number itself). A prime number is a positive integer that has exactly two divisors: 1 and itself. The conjecture states that there exists a perfect number that is also prime, which would be a very rare and interesting number.

What is the Catalan's Conjecture?

  1. A conjecture that states that the Catalan numbers are always integers.

  2. A conjecture that states that the Catalan numbers are always prime numbers.

  3. A conjecture that states that the Catalan numbers are always perfect numbers.

  4. A conjecture that states that the Catalan numbers are always odd numbers.

Show answer
Correct Option: A
Explanation:

Catalan's Conjecture is a famous unsolved problem in combinatorics. It states that the Catalan numbers are always integers. The Catalan numbers are a sequence of numbers that arise in many counting problems, such as the number of ways to parenthesize an expression or the number of ways to triangulate a convex polygon. The conjecture states that these numbers are always integers, which would have implications for many areas of mathematics.

What is the Hadwiger-Nelson Problem?

  1. A problem that asks for the minimum number of colors needed to color the vertices of a graph such that no two adjacent vertices have the same color.

  2. A problem that asks for the maximum number of colors needed to color the vertices of a graph such that no two adjacent vertices have the same color.

  3. A problem that asks for the minimum number of colors needed to color the edges of a graph such that no two adjacent edges have the same color.

  4. A problem that asks for the maximum number of colors needed to color the edges of a graph such that no two adjacent edges have the same color.

Show answer
Correct Option: A
Explanation:

The Hadwiger-Nelson Problem is a famous unsolved problem in graph theory. It asks for the minimum number of colors needed to color the vertices of a graph such that no two adjacent vertices have the same color. This problem is related to the four color theorem, which states that any map can be colored with four colors in such a way that no two adjacent regions have the same color. The Hadwiger-Nelson Problem asks whether there is a similar result for graphs.

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