I was going to suggest clear contact paper too but I think it would be difficult to place the puzzle together on it because of the stickiness and how would you know how to place it on the sticky paper 'just so' while building?
I suggest using the paper after the puzzle is together. Slide back just one edge of the paper protecting the tacky side and slowly 'press and smooth' sticky side onto one edge of the puzzle then slowly peel back more of the protective paper and continue pressing and smoothing until the top part of the puzzle is covered. Carefully turn puzzle over and do the same on the back side.
Do indeed leave at least a one or two inch overlap of contact paper on all edges so that once front and back sides are covered you can press well right along the edge of the puzzle (the smooth side of a butter knife blade works well to do this just do it gently) and then you can cut the overlap edges off.
If you can buy clear contact paper (brand name of shelf/drawer liner with adhesive), you can cut a piece about 1" bigger all around than the finished puzzle, take the backing off the contact paper, and put the puzzle together on the adhesive side. Then cover the puzzle with another piece of similar size adhesive side down. Press together firmly, sealing the edges well. One water resistant placemat.
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