# Nontransitive Identities: A New Method for Creating Nontransitive Dice

## TL;DR

This paper introduces a new method called Nontransitive Identities (NI) for characterizing and creating nontransitive dice sets, expanding understanding of their patterns and relationships across multiple dice configurations.

## Contribution

The paper presents a novel framework for describing and generating nontransitive dice sets using Nontransitive Identities, including methods for extending from 3 to 5 and 7 dice.

## Key findings

- Identified possible NI for 3, 4, and 5-sided dice
- Developed methods to generate 5 and 7 dice NI from 3 dice NI
- Discovered patterns in the frequency and morphisms of NI

## Abstract

In this article, a new method for characterizing nontransitive dice is de- scribed. This new method is then used to describe the "Nontransitive Identities" (NI) that are possible for 3 dice with 3, 4 and 5 sides each as well as for 5 dice with 3 sides each. Next, we will discuss how these NI can be used to create NI that involve more dice and/or die sides. From there, the 3 dice Nontransitive Identities (NI) will be used to produce 5 dice NI. We will begin this by describing more nomenclature that becomes necessary for characterizing sets of NI with multiple win chains. Then, we will describe a method for creating 5 dice NI and 7 dice NI from our 3 dice NI. We will conclude with a discussion on the patterns that exist in the frequency of nontransitive identities and how these patterns correlate to NI morphisms. Our ancillary files will include our python scripts as well as all of the sets of Nontransitive identities we have discussed.