This tweak of emmett's Interlocking Rings Puzzle is intended to be easier to print by using a fully-connected first layer. While this no longer has the elegance of the print-and-go puzzle, I think that's a fair tradeoff to avoid frustration with failed prints.
The original model has several tiny pieces with a very small base layer surface area which get knocked loose easily while printing. Printing with a brim seems like overkill since most of the model doesn't need it, so this model adds a small amount of brim-style connections at the intersections to keep them in place until they get bridged to the rest of the model.
Please double-check in your slicer to make sure that the connections in the first layer are present and look reasonable. The added settings are customizable:
brim_height: vertical height, adjust if needed to ensure that you get one layer of brim. If it's too thin, the slicer may ignore it.
brim_gap: a tiny gap between the support brim and model. This is intended to make the supports easier to separate cleanly.
The brim gap is supposed to ensure that the slicer preserves individual walls instead of merging adjacent volumes. (Cura respected this, Slic3r PE didn't. Is there anInterface shells settings?) Ideally lines at this separation distance should still stick together, just less strongly than the adjacent wall.
The STL has built-in brim-style connections at the intersection points. No need to add a brim or raft in the slicer.
The connections need a bit of manual cleanup. When sliced with Cura, they popped off easily for me without tools. If your slicer merged them with the bottom layer, try side cutters?