Continuity was the winner of this year's IGF student showcase (and a finalist in the 2010 Indiecade Festival). It's a puzzle platformer with the classic "get the key and then get to the exit" approach, with minimalist stick-figure graphics; the twist is that each level is broken up into several rectangular regions, which can be repositioned.
You hit space-bar to transition from platformer mode into a mode where the regions can be slid around. And you can only move from one region to another if they abut, with white areas matching across the border. For example, in the image above, you can't move from your current region to the one at the right, because the corridors are not of the same size; you need to reposition the regions so you can move into the lower one.
In reposition mode, the regions are slid around with arrow keys. It's reminiscent, in fact, of "sliding puzzles," those tacky little plastic things you get in souvenir shops that consist (typically) of a 5-by-5 grid of squares, one missing, which you have to slide around to assemble a picture.
Continuity is a good example of how two very simple and conventional ideas (platformers and sliding puzzles) can be combined in a novel way.