AMC 8 · 2003 · #13
Grade 4 geometry-3d
Pick an answer.
AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The figure is a 3D arrangement of 14 cubes, which is exactly when Tool #10 (Create a Physical Representation) earns its keep — stack real blocks (or sketch the floor plan with a separate top layer) so you can see which faces of each cube touch a neighbor. Tool #2 (Make a Systematic List) is the bookkeeping partner: group the 14 cubes by position type (bottom-corner, bottom-edge-middle, top-corner), count the neighbors for one cube in each group, and use the rule "red faces = 6 - neighbors." Three groups, three subtractions, then add the group with 4 red faces.
Build the figure: a hollow rectangular ring of 10 floor cubes plus 1 cube on each corner — 10 + 4 = 14 cubes, matching the problem.
Composing 3D shapes from unit cubes is a Grade 1 spatial skill. The split into "floor ring + corner stacks" gives a clean count and lines up the three cube types.
1.G.A.2Create A Physical RepresentationKey rule: red faces = 6 - neighbors. Since the bottom is painted too, only glued faces are hidden — the ground hides nothing.
Each touch covers exactly one face on each cube. "Bottom painted too" is the line that makes this clean — no extra subtractions for cubes sitting on the floor.
1.G.A.2Create A Physical RepresentationSort the 14 cubes by position: floor corners have 3 neighbors, floor-edge middles 2 neighbors, top corners 1 neighbor — 4 + 6 + 4 = 14.
Cubes in the same position type behave identically by symmetry, so one count per type covers all 14 cubes.
4.OA.A.3Make A Systematic ListApply 6 - neighbors: type A gets 3 red, type B gets 4 red, type C gets 5 red. Only type B hits exactly four, and there are 6 such cubes.
"Exactly 4 red faces" means "exactly 2 neighbors," which singles out the middles of the four ring sides.
4.OA.A.3Make A Systematic ListEvery face that hugs a neighbor stays white, so red faces = 6 - neighbors. Sort the 14 cubes by where they sit — floor corner, floor edge, top corner — and the 6 cubes in the middles of the ring's sides are the ones with exactly 4 red faces.