Advice on the Writing Projects for Chapter 4

1. You might start with the standard history of mathematics books, such as [Bo4] or [Ev3]. Alternatively, check some of the references on the MacTutor Web page on Dirichlet.

2. To learn about telephone numbers in North America, refer to books on telecommunications, such [Fr]. The term to look for in an index is the North American Numbering Plan. You should also consult the Web site of North American Numbering Plan Administration.

3. A lot of progress has been made recently by research mathematicians such as Herbert Wilf in finding general methods of proving essentially all true combinatorial identities, more or less mechanically. See whether you can find some of this work by looking in Mathematical Reviews or the book [PeWi]. There is also some discussion of this in [Wi2], a book on generating functions. Also, a classical book on combinatorial identities is [Ri2].

4. A general mathematics history text should cover this topic well. Introductory probability books might also have a few words on the subject. Pascal was one of the pioneers in this area.

5. It will be instructive to see whether the advice given in popular gambling books (which is where to go for this Project) is correct! Your university library might not be a good place to look for this Project; try your local bookstore instead. There are innumerable Web sites on gambling games, including those (best avoided!) at which you can actually gamble on line. Here is a site with the basic rules. You should be able to calculate the probabilities.

6. As in the previous Project, you should consult popular books on this subject. James Thorpe was one of the first persons to realize that the player can win against the house in blackjack by using the right strategy (which involves keeping track of the cards that have already been used, as well as doing the right thing in terms of drawing additional cards on each hand). The Web is full of information on blackjack (some of it probably even correct), such as the FAQ (frequently asked questions) of the "rec.gambling.blackjack" newsgroup. Of course amazon.com has many books on the subject.

7. Students who have had an advanced physics course will be at an advantage here. Maybe you have a friend who is a physics major! In any case, it should not be hard to find a fairly elementary textbook on this subject. A Web search or visit to amazon.com can help get you started.

8. More advanced combinatorics textbooks usually deal with Stirling numbers, at least in the exercises. See [Ro1], for instance. Other sources here are a chapter in [MiRo] and the amazing [GrKn]. Also try the Web site of the CRC Concise Encyclopedia of Mathematics (if the letter S is not being blocked on the day you search).

9. See the comments for Writing Project 8. Also try the Web site of the CRC Concise Encyclopedia of Mathematics (if the letter S is not being blocked on the day you search).

10. There are entire books devoted to Ramsey theory, dealing not only with the classical Ramsey numbers, but also with applications to number theory, graph theory, geometry, linear algebra, etc. For a fairly advanced such book, see [GrRo]; for a gentler introduction, see the relevant sections of [Ro1] or the chapter in [MiRo]. Also try the Web site of the CRC Concise Encyclopedia of Mathematics (if the letter R is not being blocked on the day you search).

11. Try books with titles such as "combinatorial algorithms" -- that's what methods of generating permutations are, after all. See [Ev1] or [ReNi], for example. Another fascinating source (which deals with combinatorial algorithms as well as many other topics relevant to this text) is [GrKn]. Volume 2 of the classic [Kn] should have some relevant material. There is also an older article you might want to check out, [Le1]. An interesting related problem is to generate a random permutation; this is needed, for example, when using a computer to simulate the shuffling of a deck of cards for playing card games. There is also a Web site that will generate permutations as well as many other objects.

12. See the comments for Writing Project 11.