Painted cubes

Seven unit cubes are glued together to form the solid shown. The outside of the solid is painted, but the faces resting on the table are not painted. How many of the seven cubes have exactly 4 painted faces? Try it first. Pause if you like. The question tells us there are seven cubes, including the cubes hidden in the drawing. The two stacked cubes need the middle cube underneath them. Here is the seven-cube arrangement we will examine. Lift the two stacked cubes briefly to look down on the bottom layer. Five cubes stand on the table. Two more cubes stand above the middle cube. Five plus two makes seven cubes. Set them back down. Paint reaches the outside faces. Joined faces and faces resting on the table are not painted. Separate the cubes to inspect their faces. This does not add any paint. Look at the two faces that meet at one join. Gluing the cubes together covers a face on each cube. 1 join hides 2 faces. The four cubes sticking out at the sides are the arms. An arm cube has six faces. One joins the middle cube, and one rests on the table. 6 minus 1 minus 1 is 4 painted faces. All 4 arms have 4 painted faces each. The middle cube joins all four arms. It also joins the cube above it. The table covers its last face. 6 minus 5 minus 1 is 0 painted faces. The lower stacked cube has a join above it and a join below it. 6 minus 2 is 4 painted faces. The top cube has one join. 6 minus 1 is 5 painted faces. The 4 arms and the lower stacked cube make 5 cubes with 4 painted faces. Start with six faces. Count the joined faces and remember the table.

Ten unit cubes are glued together to form the staircase shown. It is one cube deep, and its four columns are 4, 3, 2 and 1 cubes high. The outside is painted, but the faces resting on the table are not painted. How many of the ten cubes have exactly 4 painted faces? Try it first. Pause if you like. Joined faces and faces resting on the table remain unpainted here too. The staircase is one cube deep. Separate its cubes to inspect the joins. The top cube of the column three cubes high joins the taller column and the cube below it. 6 minus 2 is 4 painted faces. The top cube of the column two cubes high also has two joins. 6 minus 2 is 4 painted faces again. The single cube has one join, and its bottom face rests on the table. 6 minus 1 minus 1 is 4 painted faces. 1 plus 1 plus 1 makes 3 cubes with 4 painted faces. A face resting on the table is not painted.