Puzzle storage - "Buy some legal size poly expandable envelopes. Remove the string and replace with Velcro circles. Place in a bin. My rule is one out at a time. The envelopes are easy enough for my two year old to open and clean up himself.
Like the idea - use a board to save the walls and maybe not Velcro since it'll get gross and isn't easily cleaned. Magnets or even small hole in cup to hang on a hook? Cars, Trucks and Teething Rings: Kids Toothbrush/Bathroom Fix