Is there a one triangle to one square solution?  Yes.
This was the most frequent mistake among people sending me solutions.  Many people solved it, though.  I received solutions from David Eppstein (Geometry Junkyard), Craig Kasper, RHW320, Chris Lusby, Rozberk Omniist, Michael Reid, Greg Fredericksen (Dissections, Plane and Fancy), Nick Baxter (Sliding Block Puzzles), and Joseph DeVincentis.