Take two of the side panels. Run a bead of carpenter's glue along one of the 15-inch edges, and butt this piece against the other panel at a perpendicular angle. Make sure the edges are aligned squarely, then drive finishing nails through the side of the second panel and into the glued edge of the first panel. Use the damp cloth to wipe off any excess glue that squeezes out.
Repeat this process with the other two side panels.
Run a bead of glue along two edges of the bottom panel, then lay the panel on a flat area and butt one of the two side panel units against the glued edges. Drive nails through the side panels into the edges of the bottom panel. Wipe off the excess glue.
Repeat this process with the other side panel unit. There is only one way this unit will match up to complete the box. Run a bead of glue down the appropriate 15-inch edges and along the remaining edges of the bottom panel, then finish assembling the box with nails. Wipe off any excess glue that squeezes out, and set the box aside to dry.
Use the nailset to set the nail heads just beneath the surface of the wood, then fill the nail holes with wood putty. You can also use wood putty to fill any gaps and seams in the exposed ends of the plywood panels.
Mount the piano hinge to one end of the top panel, then attach the other side of the piano hinge to the box. This panel forms the lid of the box.
Close the lid of the box. Spread a layer of glue that completely covers the exposed top of the lid, then place the table top on it. Position it so the table top overhangs the box evenly on all sides, then place a weight on the top to "clamp" the unit until the glue dries. Wipe off any glue that squeezes out.
Use the wood putty to fill in any gaps in the edges of the table top.
Smooth all the surfaces with the sandpaper, then finish the table according to the manufacturer's instructions.