Object Type | Information | Generate |
---|---|---|
Subsets | ||
Combinations | ||
Permutations | ||
8-Queens Problem | ||
Pentominoes | ||
Permutations of a Multiset | ||
Partitions | ||
Fibonacci Sequences | ||
Magic Squares |
AMOF, the Amazing Mathematical Object Factory, shown on the left, produces lists of mathematical objects in response to customer orders. Today you are a customer and you must tell AMOF what you want produced. This factory is totally non-polluting and the objects produced are absolutely free! (There is a vicious rumour however, that the workers are underpaid and overworked.)
Click on a icon with an Indigo background, such as , to learn about that object, or on an icon with a Green background, such as to generate the object. Get it? Indigo = Information, Green = Generate; the first letters are the same! To get back to this page click on the AMOF icon , which appears both at the top and at the bottom of all succeeding pages. In addition, the SchoolNet icon appears on every page as well; clicking on it will take you back to the SchoolNet home page.
Combinatorial objects are everywhere! How many ways are there to make change for $1 using unlimited numbers of coins of all denominations? Each way is a combinatorial object known as a numerical partition. How many ways are there for Alice, Bob, and Carol to line up at the box office at a theatre? Each way is a combinatorial object known as a permutation. How many 5-card poker hands are there with a pair of aces; what is the probability of getting such a hand? To answer this question you need to know about combinatorial objects known as combinations.
But what is a combinatorial object? It's actually very difficult to define. Kind of like love: you know it when you see it, but it's hard to explain. The main feature of such objects is that there is only a finite number of any particular type. There is only a finite number of ways for persons to order themselves in line at a theatre, only a finite number of ways to make change for a dollar. On the other hand, temperature does not take on a finite number of values (it could be 25.3315411 degrees), nor does the position of a ball on a pool table; so these are not combinatorial objects. The best way to learn about combinatorial objects is to study lots of examples of them and AMOF should help you in that study.
There are many, many types of combinatorial objects and we work with just a few of them in AMOF. Some are very simple, like permutations and subsets, and some are quite complicated, like the solutions to pentomino problems.
Here is a list of some sites relevant to discrete mathematics education K-12.
The (Combinatorial) Object Server (COS) is a WWW site which has the ability to generate (i.e., list) many additional types of combinatorial objects besides those found on AMOF. COS is oriented more towards University students and researchers. However, the ideas behind AMOF (and originally, ECOS) are derived from COS.
Funding made possible through Industry Canada's |
and through the cooperation |
[Awards] | |
and the Computer Science Department of the University of Victoria. |