## 10 Sep Writing Assignments: Combinatorics

Writing Assignments:

Combinatorics:

1. Describe some of the earliest uses of the pigeonhole principle by Dirichlet and other mathematicians

2. Discuss the importance of combinatorial reasoning in gene sequencing and related problems involving

genomes.

3. Describe the different models used to model the distribution of particles in statistical mechanics,

including Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics. In each case, describe the

counting techniques used in the model.

Logic:

4. Discuss logical paradoxes, including the paradox of Epimenides the Cretan, Jourdain’s card paradox,

and the barber paradox, and how they are resolved.

5. Describe some of the techniques that have been devised to help people solve Sudoku puzzles without

the use of a computer.

6. Describe how fuzzy logic is being applied to practical applications.

Proofs and Induction:

7. Look up some of the incorrect proofs of famous open questions and open questions that were solved

since 1970 and describe the type of error made in each proof.

8. Describe the origins of mathematical induction. Who were the first people to use it and to which

problems did they apply it?

Algorithms:

9. Describe six different NP-complete problems.

10. Describe the historic trends in how quickly processors can perform operations and use these trends

to estimate how quickly processors will be able to perform operations in the next twenty years.

11. Describe what is meant by a parallel algorithm. Explain how the complexity of parallel algorithms can

be measured. Give some examples to illustrate this concept, showing how a parallel algorithm can work

more quickly than one that does not operate in parallel.

Relations and Functions:

12. Discuss the concept of a fuzzy relation. How are fuzzy relations used?

13. Describe how equivalence classes can be used to define the rational numbers as classes of pairs of

integers and how the basic arithmetic operations on rational numbers can be defined following this

approach.

Recursion:

14. Describe a variety of different applications of the Fibonacci numbers to the biological and the physical

sciences.

15. Explain the different ways in which the Encyclopedia of Integer Sequences has been found useful.

Also, describe a few of the more unusual sequences in this encyclopedia and how they arise.

16. When are the numbers of a sequence truly random numbers, and not pseudorandom? What

shortcomings have been observed in simulations and experiments in which pseudorandom numbers

have been used? What are the properties that pseudorandom numbers can have that random numbers

should not have?

Number Theory:

17. Describe the history of the Chinese remainder theorem. Describe some of the relevant problems

posed in Chinese and Hindu writings and how the Chinese remainder theorem applies to them.

18. Describe the algorithms that are actually used by modern computers to add, subtract, multiply, and

divide positive integers.

19. Show how a congruence can be used to tell the day of the week for any given date.

Graph Theory:

20. Discuss the applications of graph theory to the study of ecosystems, to sociology and to psychology.

21. Explain how graph theory can help uncover networks of criminals or terrorists by studying relevant

social and communication networks.

22. Describe some of the strategies and algorithms used to solve the traveling salesperson problem

Boolean Algebra

23. Describe some of the early machines devised to solve problems in logic, such as the Stanhope

Demonstrator, Jevons’s Logic Machine, and the Marquand Machine.

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