MOVING CHIPS AND COINS

                   Adrian Dumitrescu
                         UWM

Given a pair of start and target configurations, each consisting 
of $n$ pairwise disjoint disks in the plane, what is the minimum 
number of moves that suffice for transforming the start 
configuration into the target configuration? In one move a disk 
slides in the plane without intersecting any other disk, so that 
its center moves along an arbitrary (open) continuous curve. We 
give estimates on the number of moves under different assumptions 
on disk radii. For example, with $n$ congruent disks, ${3n\over 
2} +O(\sqrt {n\log n})$ moves always suffice for transforming the 
start configuration into the target configuration, and 
$(1+{1\over 15})n-O(\sqrt{n})$ moves are sometimes necessary.

We then compare the above {\em sliding model} with the {\em 
lifting model}, where in one move a disk is lifted from the plane 
and placed back in the plane at another location, without 
intersecting any other disk. Now $n+O(n^{2/3})$ moves always 
suffice, while $n+\Omega(n^{1/2})$ moves are sometimes necessary.

Finally, we look at what happens in the integer grid when moving 
$n$ chips to a target configuration, where each move is along 
some free path in the grid. We contrast this variant to the 
confined version: the well-known $15$ puzzle of Sam Loyd.
 
This is joint work with Sergey Bereg, Gruia C\u{a}linescu and 
J\'anos Pach.