IDEA: put the letters on the sides of the hexagon so that I don't have to think so hard.

This is a great idea!

Drunk 2048 clone idea - as time goes on probability your move will be replaced with a random move grows.

For example:

after 2 seconds: [0, 1, 0, 0]* => [0.1, 0.8, 0.1, 0]

after 4 seconds: [0, 1, 0, 0] => [0.15, 0.6, 0.15, 0.05]

after 8 seconds: [0, 1, 0, 0] => [0.25, 0.25, 0.25, 0.25]

* Probability of [left, up, right, down] move.

I think having buttons to click on the screen (or little arrows next to each side) would help a lot here.

Absolutely. There's no reason a keyboard should be necessary to play. I'd much rather use a mouse.

After playing this for what feels like an hour (and what may have actually been an hour) I'm suddenly noticing square edges a lot more in my surroundings.

Suddenly the tabs in Chrome look extremely rectangular...

holy crap! I just played for two hours ( got to 2048 ) and when i was done everything (win7) looked very square, kind of small and ... old-fashioned? holy crap. It got back to normal after about five minutes or so...

I genuinely thought something had gone wrong with my browser. It's surreal.

I had that too! Switching back to my IDE I was surprised at how "boxy" my code suddenly looked. It's a really weird psychological phenomenon. I wish I knew why it was happening.