Rock, Paper, Scissors, Spider, Man

Have you ever gotten bored of the classic rock, paper, scissors game? Fear no more! Here is a new twist on the classic that will keep you entertained for hours!

# The Rules

## Spider

### The spider wins against:

**Rock**because spiders have been known to withstand the weight of rocks (evolution is amazing)**Human**as humans are terrified of spiders

### The spider loses against:

**Paper**because spiders can be captured by paper**Scissors**because scissors can cut spider webs (and spiders)

## Man

### The man wins against:

**Paper**because humans can shoot paper with guns**Scissors**because humans can shoot scissors with guns

### The man loses against:

**Rock**because humans can be crushed by rocks**Spider**because humans are terrified of spiders

## Rock

### The rock wins against:

**Scissors**because rocks can crush scissors**Human**because rocks can crush humans

### The rock loses against:

**Paper**because rocks can be covered by paper**Spider**because spiders can withstand the weight of rocks (evolution is amazing)

## Paper

### The paper wins against:

**Rock**because paper can cover rocks**Spider**because spiders can be captured by paper

### The paper loses against:

**Scissors**because scissors can cut paper**Human**because humans can shoot paper with guns

## Scissors

### The scissors wins against:

**Paper**because scissors can cut paper**Spider**because scissors can cut spider webs (and spiders)

### The scissors loses against:

**Rock**because rocks can crush scissors**Human**because humans can shoot scissors with guns

## Cayley Table

Rock | Paper | Scissors | Spider | Man | |
---|---|---|---|---|---|

Rock | Draw | Paper | Rock | Spider | Rock |

Paper | Paper | Draw | Scissors | Paper | Man |

Scissors | Rock | Scissors | Draw | Scissors | Man |

Spider | Spider | Paper | Scissors | Draw | Spider |

Man | Rock | Man | Man | Spider | Draw |

## On the existence of n-dimensional generalizations

Denote the game as $\mathbf{G}$ with binary operation $\circ$ and identity element $I$.

Given a set $\mathbf{S}$ of $n$ elements ($a_i$), we can define a Cayley table for $\mathbf{G}$ as follows:

Call the game $\mathbf{G}$ *balanced* if $|\{a_i \ | \ a_i \circ a_j = a_i, \ \forall \ 1 \leq j \leq n, \ \forall \ a_i \in \mathbf{S}\}| = \frac{n-1}{2}$

Indeed for a $\mathbf{G}$ to be nontrivial it must satisfy:

- $\mathbf{G}$ is closed under $\circ$ (by definition)
- $\mathbf{G}$ is balanced

*Claim*. A balanced game $\mathbf{G}$ exists if and only if $n$ is odd.

*Proof*.

Disincluding, the identity element, there are $n-1$ elements in $\mathbf{S}$. For $\mathbf{G}$ to be balanced, there must be $\frac{n-1}{2}$ elements that are their own inverse. This is only possible if $n-1$ is even, i.e. $n$ is odd.