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Cut the plywood to size: two pieces 36" x 18"; two pieces 18" x 18", and one piece 38" x 20". Cut the masonite to 36" x 18". Rout a groove along the bottom edge of each piece of plywood to hold the masonite.
Assemble the toybox by gluing the edges together and gluing the masonite into the grooves. Then nail the pieces together with finish nails. Cut some blocks from scrap lumber and fit them into the corners to provide extra bracing. The largest piece of plywood is for the lid. The overlap provides a place to grab it. Attach it to the box with a piano hinge.
Install a lid closer, which will ensure that the lid closes slowly, preventing smashed fingers and bumped heads. You can buy lid closers at hardware stores.
To decorate the box, cut designs out of 1/4" plywood. You can use a saber saw clamped upside down to a scroll-saw table. First draw the designs on the plywood, then just move the wood around so that the blade follows the outline of the shape. Once they're sanded and painted, you're ready to glue them to the box.