Collection of conjectures that are easy to understand

Showing students these problems reinforces the idea that mathematics is not done and dusted. Numberphile offers videos on all of these problems. You could have students play with these problems and prove them for special cases. For example, it's pretty much obvious that the Collatz conjecture holds for powers of 2. What other sets does it hold for? Similar results and questions can be formed for the other conjectures as well.