Mod Cycle
Starting with 3, let’s double it and add one (or 2n+1), then modulate the answer by 14 (divide by 14 and take the remainder). If we repeat these two steps, we get the first cycle.
Jumping Tiles
In the following puzzle, each number indicates the distance (vertically, horizontally, or a combination of both) this tile needs to move. A solution is a rearrangement of all the tiles in the grid that satisfies all the numbers where none of the tiles overlap.