A variety of ways that Fibonacci number arise in nature, including counting rabbits, can be found on Ron Knott’s page at the Department of Computing, University of Surrey site:

To go straight to the rabbits, see

Page 311

A picture of the 19^th century original box cover for the Tower of Hanoi puzzle and the text of the original instructions in French, and translated into English, can be seen at the site of Paul K. Stockmeyer, a professor of computer science at William and Mary College.

Several interesting papers about the Tower of Hanoi problem and its generalizations written by Paul K. Stockmeyer can be downloaded from

Many Web sites run demonstrations of the Tower of Hanoi puzzle. Some of these demonstrate the moves used to solve the puzzle in an animation. Others run in an interactive mode where you can move disks yourself (by clicking and/or dragging) . Here are some of these sites. (Note: For some of these sites you must have a Java-enabled browser to run the animated and/or interactive programs.)

A cleverly constructured page, written by Glenn Crocker, updates the legend of the monks moving the 64 gold disks. In this version, a monk is shocked into moving a disk whenever someone visits the web page that displays the current status of the disks. When the tower of 64 disks has been moved, this page tells us that the internet will come to a crashing halt! The current status of the monks progress is also noted.

Some history of the Tower of Hanoi problem, animation, and interactive software are available at the Lawrence Hall of Science site:

Other interactive version of the Tower of Hanoi can be found at

A program designed to study generalizations of the Tower of Hanoi puzzle with more than three pegs can be found at the Multipeg Tower of Hanoi site. The program, which runs in different levels, can handle from 1 to 500 disks on from 3 to 30 pegs. In level 1, it demonstrates the algorithm presumed to be optimal (the Frame-Stewart algorithm) for the generalized Tower of Hanoi problem with more than 3 pegs. In levels 2 and 3 the user can compete against the computer. In level 4, the user can solve the puzzle by clicking on the pegs. It was written by Xue-Miao Lu, a researcher on variations of the Tower of Hanoi .

A biography and a photo of Catalan can be found at the MacTutor History of Mathematics Archive at the University of St Andrews, Scotland.

A list of 17 different contexts in which the Catalan numbers arise can be found at

An attractive page, part of Robert M. Dickau’s Mathematical Figures site, provides illustrations of many of the objects that Catalan numbers count,

The United States Census Bureau provides a variety of information about population and population growth for the United States and for the world. For information about the current population of the world consult

The growth rate of the population of the world is actually projected to decrease. For the latest view, consult

The solution of homogeneous and nonhomogeneous recurrence relations is covered in extensive lecture notes written by Zhuhan Jiang at the University of New England in Armidale, Australia.

Paul E. Dunne of the Department of Computer Science at the University of Liverpool, England, has written an excellent introduction to divide-and-conquer algorithms, including discussions of the binary search, multiplication of integers, and finding the closest pair of points in the plane. This site can be found at

Material on probability generating functions can be found in the Expected Values section of the Virtual Laboratories in Probability and Statistics site.

Biographical information about Eratosthenes and a demonstration and applet of the sieve of Eratosthenes can be found at

A biographical and portrait of Eratosthenes can be found at

An interesting way to look at derangements graphically can be found at Robert M. Dichau’s Mathematical Figures site:

All the derangements of n objects can be generated using an interactive program at the Combinatorial Object Server

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