Cracking the Humpty Dumpty Case

A favorite puzzle in game stores is a wooden T that has been cut 
into four pieces.  While it seems a simple task to reconstruct 
the T, many people struggle fruitlessly, and eventually give up.
Puzzles like this are called geometric dissection problems.
Computers can work on these problems, but the usual approach
is an unimaginative example of ``brute force'', that is, just
try every possibility until you stumble on the answer.  In this 
talk, I will show how to create a corresponding tiling problem, 
also known as an exact cover.  This results in an underdetermined 
integer linear programming problem of the form Ax=b.  Fast solvers 
can handle such problems, involving hundreds of pieces, and 
so we can put Humpty Dumpty together again.