The Art and Craft of Problem Solving

References

REFERENCES AND FURTHER READING 357

text [15] is a fine alternative. For more advanced reading, there are three books that

truly stand out: Concrete Mathematics [16], generatingfunctionology [47], and Proofs

that Really Count: The Art of Combinatorial Proof [4]. The first is an amazingly

readable and comprehensive text (with full solutions), the second is a poetic guide

to generating functions (and much more, also with full solutions), and the third is an

amazingly imaginative exploration of the question: How far can we go by recasting

combinatorics with pictures?

There is also no shortage of excellent number theory books. However, none of

them stand out as must-reads. We recommend Vanden Eynden [45] for beginners,

and Niven and Zuckerman [30] or Ireland and Rosen [23] for more advanced readers,

although we are sure that there are other worthwhile choices.

For geometry, we hope that our book has prepared you to read the excellent Ge­

ometry Revisited [6]. For more comprehensive coverage, the survey by Eves [10] is

unsurpassed. Needham's Visual Complex Analysis [29], besides its many other trea­

sures, contains a beautiful treatment of transformations and non-Euclidean geometry,

from a very interesting perspective. Our favorite source of geometry problems (with

full solutions) is Prasolov's Problems in Plane Geometry [35]. Unfortunately, it is in

Russian. However, we hope to translate this wonderful resource soon!

As with combinatorics, we recommend three classic advanced calculus texts that

stand head and shoulders above the rest: the books by Boas [5], Spivak [39], and

Apostol [2]. The book by Boas stands out in particular, because it is less than 200

pages long!

1. Gerald L. Alexanderson, Leonard F. Klosinski, and Loren C. Larson, editors. The
   William Lowell Putnam Mathematical Competition. The Mathematical Associa­
   tion of America, 1985.


2. Tom M. Apostol. Calculus. Blaisdell, second edition, 1967-69.


3. E. J. Barbeau. Polynomials. Springer, 2003.


4. Arthur T. Benjamin and Jennifer Quinn. Proofs that Really Count: The Art of
   Combinatorial Proof The Mathematical Association of America, 2003.


5. Ralph Boas. A Primer of Real Functions. The Mathematical Association of Amer­
   ica, second edition, 1972.


6. H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited. The Mathematical
   Association of America, 1967.


7. N. G. de Bruijn. Filling boxes with bricks. American Mathematical Monthly,
   76:37-40, 1969.


8. Ilia Itenberg Dmitri Fomin, Sergey Genkin. Mathematical Circles (R ussian Expe­
   rience). American Mathematical Society, 1996.


9. Heinrich Dorrie. 100 Great Problems of Elementary Mathematics. Dover, 1965.


10. Howard Eves. A Survey of Geometry, volume 1. Allyn and Bacon, 1963.


11. Martin Gardner. The Unexpected Hanging. Simon and Schuster, 1969.