# Uncovering the Secret of Nim

Hard
Have you ever heard of the game of Nim? It's a mathematical game that involves binary addition without carrying. We'll explore the properties of Nim addition and how it relates to multidimensional objects called Simplexes.

## Hypothesis

The hypothesis is that there is a mapping between Nimbers (non-negative integers under Nim addition) and Simplexes.

## Method & Materials

You will research the game of Nim, Nimbers, Nim addition, group properties, and the winning strategy for Nim. You will also research Simplexes and their properties. You will prove that Nim addition has the properties of an Abelian group and show how Nimbers can be counted with their base 2 representations. You will determine a way to find the number of vertices, edges, and faces in a Simplex and prove that there is a one-to-one mapping between Nimbers and Simplexes. You will explore 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. You will construct a Simplex-3 using Zometool to illustrate this mapping.
You will need Zometool, a Dell PC running Microsoft Windows 98 and Word 97, and an HP printer.

## Results

This project has demonstrated that Nim addition has the 5 Abelian group properties and that Nimbers can be graphed onto Simplexes for lines, triangles, and tetrahedrons. It has also been proven that there is a one-to-one mapping between Nimbers and Simplexes, and that Nimbers determine their own unique Abelian groups and are locations in multidimensional space.

## Why do this project?

This project is so interesting and unique because it shows how finite groups can relate very different mathematical objects to each other.

## Also Consider

Experiment variations to consider include exploring the properties of the mapping between Nimbers and Simplexes in more detail, and constructing a Simplex-4 to illustrate the mapping.

## Full project details

You can find additional information and details for this science fair project here. Have fun exploring!