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Title: The Secret of Nim: Mapping Finite Groups under Nim Addition to
N-Dimensional Simplexes


The purpose of this project is to determine if there is a mapping between
Nimbers (non-negative integers
under Nim addition) and multidimensional objects called Simplexes, thus
demonstrating the power of
finite groups to relate very different mathematical objects to each other.


I researched the impartial mathematical game of Nim including Nimbers, the
basics of Nim addition
(binary addition without carrying), group properties, and the winning strategy
for Nim as proven by
Charles L. Bouton. I also researched Simplexes and their properties. I proved
that Nim addition has the
properties of an Abelian group and showed how Nimbers can be counted with their
base 2 representations.
I also determined a way to find the number of vertices, edges, and face-ns in a
Simplex and proved that
there is a one-to-one mapping between Nimbers and Simplexes. I then explored
properties of this mapping
to show how a Simplex could be used for Nim addition and how Nimbers determine
their own unique
Abelian groups and are locations in multidimensional space. I constructed a
Simplex-3 using Zometool to
illustrate this mapping. Materials used in this project are Zometool, a Dell PC
running Microsoft
Windows 98 and Word 97 and an HP printer.


This project describes Nim addition as binary addition without carrying and
shows how to use Nim
addition to win a game of Nim. It demonstrates that Nim addition has the 5
Abelian group properties.
Graphing Nimbers onto Simplexes is shown for lines, triangles, and tetrahedrons.
The mapping of
Nimbers to n-dimensional Simplexes is proved and illustrated using Pascal's


The hypothesis that there is a mapping between Nimbers and Simplexes was proven.
This required an
analysis of Nim addition with proofs of its Abelian group properties.
Combinatorics, one-to-one
mappings, Pascal's triangle, and the binomial theorem were all utilized for the
proof. Other observations
include how to use the graph for Nim addition, and the fact that Nimbers are
locations in
multidimensional space and determine their own unique Abelian group.

Summary Statement

This project maps finite Abelian groups under Nim addition (binary addition
without carrying) to
multidimensional objects called Simplexes, demonstrating their use to win the
game of Nim.

Help Received

My dad helped me research difficult topics and taught me how to set up a
backboard. My mom made sure
that the project could be understood and helped with the backboard layout. Mr.
Sewell, my math teacher,
suggested various corrections to my report.