The Rule of Two is not just about Pairs

Okay, last time I set up an abbreviated array that looked like this:

K|K|  
  |K|K
  |  |K       

with givens in TL, TC, MC, MR, and BR, and then said that I knew I'd immediately be able to place a deduction in the Top Right square.  That's because the two givens in the Top zone will limit the possible spaces in the array for our mystery symbol in TR to one row, while the two givens in the Right zone will limit the spaces in TR to one column.

Similarly, I can assert that I'll find pairs in ML and BC because in each case one of the two intersecting zones has two givens, bringing us down to one segment, and the other intersecting zone has one given, which will take away one of the spaces in that segment.

BL, with only two givens influencing it, will have to wait for more data.  Of course, the abbreviated array is only looking at one symbol, and there may be different symbols also within that square which will limit the possible spaces and get us an answer or a pair.  But abbreviated arrays can be a quick shorthand way of looking for the places where you'll get the most bang for your buck.

Of course, most of the time you won't have five givens in your array to start with.  Let's take a look at what happens with three givens in some different arrangements.

K|K|K
  |  |  
  |  |                

K|K|  
  |  |  
  |  |K  

K|K|  
  |K|  
  |  |  

In the first example, we're dependent on how crowded all the other squares are with other symbols.  There's nothing we can learn from the abbreviated array.

In the second example we know we'll find at least one pair in TR.  

In the third example we can't guarantee finding anything, but it will take a lot less crowding in TR, ML, and BC from the other arrays to produce results.

Now we're ready to definitely start thinking Too Much.