Some interactions between crowdedness and influences.

Let's start with one of the simplest sets of interactions.  We'll look at squares with three spaces left and a single influence.

X	X	X		t					t
X	X	X		m					m
				b					b
l	c	r
l	c	r

For each influence, I've color coded the result.

Black is no effect

Green is pair

Blue is deduction

Red is impossible/invalid

So in this first chart we can see that out of six occasions, three have a result and one can never happen in the game.  So 3/5ths or 60% of the time we get at least a pair.

But of course, the "crowd" in the square mighte be in a different configuration.

How about something that looks like this:

X	X	X		t					t
X	X			m					m
X				b					b
l	c	r
l	c	r

Here, our results are different.  There are no invalid interactions, and while two of the interactions have no result, two of them (b) will give us a hit, not just a pair.  If 2/3rds of our interactions give us a result here, that's 66%.  

We can create different charts with three spaces, but are they functionally equivalent to this one?

Nothing to do but test it.

X	X			t					t
X	X			m					m
X		X		b					b
l	c	r
l	c	r

And clearly the answer is no.  We're looking at one result with no influence, but we've raised our pairs to four and reduced our hits to one.  

5/6ths into percentages is more than my brain is up to just now.  But I've got one more chart for us to investigate.

This time, we'll break things up a bit.

X		X		t					t
X	X			m					m
X		X		b					b
l	c	r
l	c	r

And we get a similar result.  One dud, one hit, and four pairs.

What makes this arrangement functionally equivalent to the one above it?

Something to think about.