Why would my dog start eating out of only one side of her bowl?

She's a 12 year old border collie mix, had her teeth cleaned 2 months ago. Otherwise is acting fine, this started 2 days ago 

This is a fascinating presentation.  Of course, I couldn't tell you what is happening in your individual dog's case because there are various possibilities and you will need to see your vet, perhaps for a full examination and a neuro exam, to establish which seem most likely.  Tests may be needed from there.  Is it always the same side that she doesn't eat from?  I wondered about pain eg neck or back pain.  Dental pain perhaps. But then, most dogs with dental pain simply eat all the food with the other side their mouth, rather than the pattern you are describing - and dogs with neck pain might move around the bowl. Another group of possibilities involves changes to your dogs' ability to detect food in that area of the bowl; changes to smell or vision for example, perhaps arising from the sensory organs themselves (although dogs with poor olfaction normally see food and dogs with poor eyesight, smell it) or arising from a lack of perception within the brain.  If your vet is suspicious of this, they might suggest an MRI scan or neirological referral.   On the other hand, it may simply be possible that your dog has tummy ache eg pancreatitis, becomes full after a while and is a methodical eater, so this is what the bowl looks like when they run out of steam.  I would suggest allowing your vet to take a full history, examine your dog and to suggest investigations from there based on the results.

Thank ypu for sharing this interesting case; please will you let us know what happens next?  Your vet should triage the case in order to decide how urgently your pet needs to be seen.