There are lots of Social Media games out there so I thought, why not have your residents compete with each other online? I think this would be really easy to get going...but you may need someone familiar with both computers and smart phones to help teach your residents how to take and send screen shots.
So, invite everyone who plays something like "Words with Friends" to send you their user names if they would like to participate in a tournament... they don't have to share the rest of their identity with the community if they don't want to but you will need to know if you are to distribute any prizes.
Put all the usernames into brackets and post it somewhere. Make it clear who is to start the game so that they're not both inviting each other at the same time. Something like "cutiejill invites smartypants" Then, let them play out their game. At the end of the game, when someone is declared a winner, have them take a screen shot and send it to you so you can update the board and get the next match started.
I really think this is a great way to get your community together for today's version of a Board Game Night. And you'll get some old pros who help make the process go much more smoothly.