Algebra is a branch of math that solves problems using letters and symbols instead of just numbers. Using letters helps us to represent numbers that we don't know the value of yet, could change, or to generalize formulas or equations.  

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.

Does a simple number game hold a hidden link to shapes in higher dimensions? The game of Nim uses binary addition without carrying. The numbers in this system are called Nimbers.

You prove that Nimbers form an Abelian group (a set where addition follows certain rules). Then you map each Nimber to a point on a Simplex (a shape like a triangle or pyramid extended into more dimensions). You build a 3D Simplex model to see this mapping in action.

The mapping turns out to be one-to-one. Each Nimber matches exactly one point in multi-dimensional space. You can even use the Simplex graph to perform Nim addition visually.

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

Objectives/Goals

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.

Methods/Materials

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.

Results

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
triangle.

Conclusions/Discussion

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.

Nim Addition and Simplex Geometry

Mapping Finite Groups under Nim Addition to N-Dimensional Simplexes

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Nim Addition and Simplex Geometry

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