OHR has a number of fascinating questions and answers on what is and isn’t permissible on Shabbat.
What you can and can’t do:
Velcro – is it tearing?
Lifts (elevators) – there are special Shabbat lifts
Glow in the dark toys – is moving electrons allowed?
Motion Sensors – as long as you’re careful
Scrabble, Monopoly & Chess – Chess but no Chess clocks
Emailing across timezones into the Shabbat – depends on the server
Origami – nope
A light dilemma – special automatic lights are available
[Partially via Waxy]