# solitaire

and topology

2012–09–28

Do they play nicely?

Do they talk to each other?

And if they do, what do they say to each other?

## basics

Warning: this is rubbish. Rubbish written by an idiot who thinks that he knows more than he does. I might have something here but please do not treat what follows as in any way true.

Can we create a topological space that represents a solitaire board? The answer is yes, in fact we can create many. What we need is a useful one, or a useful several perhaps. I’m going to create[?] one that has some interesting properties and some interesting oddities.

You can’t play solitaire seriously for long before you notice the *classes*. Let’s have a look at them [for the English board, which I’m going to restrict this analysis to for now]. In a pretence at mathimatical rigour, let `X` be the union of all the filled holes of a solitaire board, and let `A, B, C` and `D` be proper subsets of `X`. Let’s have a look at these `A, B, C and D`’s, shall we?.

0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

0 | - | - | - | - | |||

1 | - | - | • | - | - | ||

2 | |||||||

3 | • | • | • | ||||

2 | |||||||

5 | - | - | • | - | - | ||

6 | - | - | - | - |

0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

0 | - | - | • | • | - | - | |

1 | - | - | - | - | |||

2 | • | • | • | • | |||

3 | |||||||

4 | • | • | • | • | |||

5 | - | - | - | - | |||

6 | - | - | • | • | - | - |

0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

0 | - | - | • | - | - | ||

1 | - | - | - | - | |||

2 | • | • | • | ||||

3 | |||||||

4 | • | • | • | ||||

5 | - | - | - | - | |||

6 | - | - | • | - | - |

0 | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

0 | - | - | - | - | |||

1 | - | - | • | • | - | - | |

2 | |||||||

3 | • | • | • | • | |||

4 | |||||||

5 | - | - | • | • | - | - | |

6 | - | - | - | - |

The *basic* point is that no peg can move from one class to another, and it has to interact with another class to be able to move. [And A, B, C and D are disjoint and their union is X.] If we are going to discuss solitaire topologicaly it would seem to make sense that these are open sets. These alone aren’t a topological space, we need to include all the unions. I won’t write this down, it’s a horrible thing to look at and doesn’t get us any further forward. Let’s just call this topology `Τ ^{S}`

#### the idea

For now is that *if* we can find an iterated function that ends with a connected courser-topolog&yyellip;?

It’s a wee bit tough, to find such a function on a solitaire board because we’ve never defined what a function is on a solitaire board. We are going to have to be very mad here.