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 mathematical 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 topologically 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

Let’s notice that X is disconnected.

the idea

For now is that if we can find an iterated function that ends with a connected courser-topology…? A single point?

However before we tackle that we have something tricky to think about—we want this to be a metricizable topology, what is a distance on a solitaire board? We’ll tackle that next time.