OK, so a black toddler and a Chinese toddler stumble on an international drug-trafficking ring — no, actually, this is a game I just bought for CJ, a kid’s version of Nob Yoshigahara‘s classic game Rush Hour. The object here is to get the small white truck to the edge of the board (the top edge, in the image here.) The trucks in your way can’t move sideways or turn; they just go forward and back.
You play a captivating game like this and naturally you start abstracting out the underlying math problem. Play Set enough and you can’t avoid thinking about affine caps. Rush Hour has more to do with the geometry of configuration spaces; it reminds me of the “disk in a box” problems that people like Persi Diaconis and Matt Kahle work on.
So here’s a question — it doesn’t capture all the features of Rush Hour, but let’s start here. Let X be the unit square, and let c be a parameter between 0 and 1, and let N be a large integer. Let L be the set of line segments in X which are either horizontal of the form y = i/N or vertical of the form x = i/N. A traffic jam is a choice of a length-c interval in each of the 2N +2 line segments in L, where we require that these intervals be pairwise disjoint. The traffic jams naturally form a topological space, which we call T(N,c). We say an interval (x,i/n),(x+c,i/n) in a traffic jam t is trapped if no traffic jam in the connected component of t contains the interval (0,i/n),(c,i/n).
Questions: For which values of (N,c) is T(N,c) connected? In particular, is it connected almost always once it’s nonempty? If not, when does T(N,c) have a “giant component”? If there’s an interesting range of parameters where T(N,c) is not connected, what proportion of intervals do we expect to be trapped?