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?.

Class A
 0123456
0--   --
1-- • --
2       
3 • • • 
2       
5-- • --
6--   --
Class B
 0123456
0--• •--
1--   --
2• • • •
3       
4• • • •
5--   --
6--• •--
Class C
 0123456
0-- • --
1--   --
2 • • • 
3       
4 • • • 
5--   --
6-- • --
Class D
 0123456
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 ΤS

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.