## The hardest Rush Hour position

It takes 93 moves to solve, per this paper by Collette, Raskin, and Servais.  I tried it and got nowhere.

You can think of the space of all possible configurations of vehicles as, well, a configuration space, not unlike the configuration spaces of disks in a box.  But here there is a bit less topology; the space is just a graph, with two configurations made adjacent if one can be reached from the other by making a single move.  The connected component of configuration space containing the “hardest case” shown here has 24,132 vertices.

I wonder what this graph looks like?   What does the path of the cars look like as you traverse the 93-step path; do most of the cars traverse most of their range?  How many of the possible configurations of the 13 vehicles (constrained to stay in the given rows and columns, and in the same linear order when two share a row or column) are actually contained in this component?  Maybe Matt Kahle knows.  By the way, another Matt Kahle-like fact is that among the list of the hardest configurations are some which are not so dense at all, like this one with only 9 cars.  It looks like it should be easy, but apparently it takes 83 moves to solve!

## Rush Hour, Jr.

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 capsRush 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?