Inspired by stories of strangely behaved Christmas lights here on Puzzling, I feel comfortable now to share my own Christmas light horror story.

It all started when I ordered a grid of star lights on eBay. Advertised as 'inifinitely bright!', I wasn't sure what to expect when I unboxed them. To my surprise, the grid of stars was infinitely large! More to my surprise, a finite number of the stars were OFF.

Obviously, I couldn't deal with such a tacky detail. I tried turning them all on but only some of them would turn on; confused, I read the instructions:

'You may only switch a light on/off if it is adjacent to an EVEN number of lights that are on (including zero).'

What a fuss! I eventually managed to turn the lights all on. However, I then realised I didn't have any place to fit an infinitely large wall of lights, so I put it back into the box and asked for a refund from the eBay seller, who only sent me a fez and the apology 'take this instead, fezzes are cool.'

If there's a silver lining to this, it's that I started to ponder a strange puzzle:

Suppose you had an infinite grid of lights, of which a finite number are off and the rest are on, and you can only toggle a light if it has en even number of orthogonally adjacent ON lights.
Can you characterise ALL light patterns that can get completely turned on?

(EXAMPLE: Suppose only 4 lights were off, in the following pattern: 

```
  1  
 2  
34  

```

You can't toggle lights 2 or 3. But you can turn 1 and 4 on, and then turn 2 and 3 on.  

This is an easy example, some patterns of lights will require you to turn other lights OFF before you can make useful progress.)

EDIT: This is something known as a 'combinatorics' problem in Olympiad Mathamatics. Cominatorics problems are notorious for having many 'fake solves', or solutions that seem solid but actually skate over important parts of the question. Both answers so far are definitely on the right track but are unfortunately not complete solutions. It's super easy to make assumptions and statements that don't always hold... be careful!