Competition · AMC preparation · step 4 of 4
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 block figure
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 RepresentationSet the red-face rule
Key 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.
A cube's number of red faces equals 6 minus the number of neighboring cubes it touches.
▸ Why?
The 6 faces of a cube split cleanly into the red ones and the white ones, so the two counts add back to 6, and taking the white count away from 6 leaves the red count.
▸ Why?
The red faces and the white faces are two non-overlapping groups that together use up every face, so their counts add back to the full 6.
▸ Why?
Since the red count plus the white count is 6, undoing that addition by subtraction gives the red count as 6 minus the white count.
▸ Why?
The white faces are exactly the ones a neighbor covers, and those match the neighbors one for one, so the white count equals the number of neighbors.
▸ Why?
A face stays white exactly when a neighboring cube is pressed flat against it, because the whole outside surface including the bottom was painted, so only the sealed contact faces escape the paint.
▸ Why?
Each neighbor meets this cube along exactly one shared face, and two different neighbors press on two different faces, so neighbors and hidden faces pair up one for one.
Sort the cubes by neighbors
Sort 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 the rule to each type
Apply 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.
- Build the block figure
- Set the red-face rule
- Sort the cubes by neighbors
- Apply the rule to each type
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