The Collatz conjecture — take any positive integer, if even divide by two, if odd multiply by three and add one, and you always eventually reach one — has resisted proof since 1937 despite its elementary statement. A new preprint from a mathematician at the University of Bonn doesn't prove it, but the density arguments introduced to show that almost all integers eventually descend may represent the most substantial methodological progress on the problem in decades. The mathematical community is cautiously interested.