You are walking down one side of a hallway. It is only wide enough for two people to pass side by side.
Due to shoddy logic puzzle-appropriate workmanship, the hallway lights only work when nobody in the hallway is moving, and otherwise it is complete darkness.
A person is coming in the opposite direction, on the same side as you.
As a logician you are far too shy to be able to speak to this other person directly.
How do you avoid colliding with them?
Assuming they are logical as well, both of you know you can't keep walking forward in the dark without risking bumping. If you change sides, they could as well. So you both stop waking and the lights turn on.
Because you both know time isn't discrete, if the lights go out and you weren't moving, then it was your partner that was moving. The person who initiates the darkness gets to move. Wait a random amount of time and if the lights didn't go out, swap sides of the hallway.
Once the swap is done (by the one who moved first) and you both see you are no longer going to collide, both resume waking forward.