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Step-by-step build — 6 steps. Also available as a print sheet.
Screw the 2x4 sides and ends together into a rectangle. Check it's square by measuring the two diagonals — when they match, it's square. Easy peasy, but don't skip it: this shape is what every later piece registers against.
Attach a leg block to each corner, then screw the leg to it. 2x4 works fine in place of 2x3 for the blocks. Taper the legs or add decorative feet here if you want — it's purely cosmetic at this stage.
Attach cleats along the inside of the box to carry the mattress slats, using predrilled holes and 2-1/2" screws.
Glue and screw the 1x3 lip pieces around the top outside edge of the box on the head and side rails, standing proud of the box top — this raised edge is what gives a platform bed its look and keeps the mattress from sliding off sideways.
Screw the footboard lip piece to the bottom rail with 2-1/2" screws and glue, completing the raised edge around the box.
Lay the slats across the cleats, evenly spaced, and screw each one down so it can't shift. A firmer mattress can get away with fewer, wider-spaced slats; a softer one needs them closer together.