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One of my friends recommended spherical magnets after I showed him my first batch, and I was surprised you could still get the things XD, but I can easily modify the design to accommodate them. I think my pieces had 6mm from the cups to the edge, I had just started designing for the magnets i had around the house.

My initial design had exposed magnets and would have had the coaster just glued on, and my legs just inserted into sockets in the bottom, but mostly I wasn't satisfied with the bridged portion even though it would have been covered by a coaster (and really didn't think they looked good enough to try letting it bridge the cup areas). The legs aren't even absolutely necessary, and could be removed entirely if you didn't need them to stay as nicely stacked. (mostly I end up designing things to use as few supported bits as possible, which usually ends up with a few more parts to be assembled)

I used the cork mainly so that the coaster would be absorbent, but it could easily be replaced by an infill disc, maybe with just the roof turned off so that there was still a bottom to the area so any condensation could be contained... Coasters should dry enough in most regions that I wouldn't expect anything to grow in them between uses.