Competition · AMC preparation · step 4 of 4
AMC 8 · 2019 · #12
Grade 0 geometry-3dlogic
Pick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a 3D cube problem, so Tool #17 (Visualize Spatial Relationships) is the natural fit — we need to picture how the faces sit around the cube. A real die or a paper cube (Tool #10) makes the adjacency facts undeniable for a young learner. Notice that Red appears in all three views, so Red is the perfect "pivot" face: by collecting the neighbors of Red across the three views, we can build the full list of faces adjacent to Red. The clever move is then Tool #16 (Change Focus / Complement): instead of trying to locate Aqua directly, we find the face opposite Red (the one missing from Red's neighbor list), and then use the symmetry of "opposite" to flip the question around.
Recall the cube's face setup
Every face of a cube has exactly 4 neighbors (sharing an edge) and 1 opposite face across the cube.
Knowing that every face of a cube has exactly 4 neighbors and 1 opposite is just basic 3D-shape knowledge from Kindergarten geometry.
K.G.B.4Visualize Spatial RelationshipsRead the three views
At each corner the three visible faces are pairwise adjacent, so every view hands you a batch of "X touches Y" facts.
Holding a real block and matching each picture to a corner makes the adjacency facts obvious by touch.
K.G.B.4Create A Physical RepresentationCollect Red's neighbors
Use Red as the pivot: across the three views, Red touches Brown, Green, White, and Purple.
Combining the neighbor sets from the three corners builds the full "around Red" picture from small pieces.
K.G.B.4Visualize Spatial RelationshipsEliminate to find the opposite
Red already claims itself plus those 4 neighbors, so the only leftover color, Aqua, must be the face opposite Red.
Out of 6 colors, 5 are accounted for, so the 1 left over (Aqua) must be the opposite face — a Kindergarten "missing addend" idea.
The face opposite the Red face is Aqua.
▸ Why?
Each of the six colors plays exactly one role toward Red — it is Red itself, one of Red's four neighbors, or Red's single opposite — so the one color that is never a neighbor of Red is forced into the opposite spot, and that color is Aqua.
▸ Why?
A cube's six faces divide with no gaps or overlaps into Red, Red's four neighbors, and Red's one opposite; taking away Red together with its four neighbors leaves exactly one color, which must be the opposite face.
▸ Why?
Red's four neighbors are Brown, Green, White, and Purple, gathered from the three views, so Aqua is the only color that never sits next to Red.
▸ Why?
In each view the three visible faces belong to one corner of the cube, and those faces occupy neighbor roles around Red rather than the far opposite role, so every face pictured beside Red is a neighbor and their colors are Brown, Green, White, and Purple.
Flip the relation
"Opposite" is mutual: since Aqua sits opposite Red, the face opposite Aqua is Red — choice (A).
On a cube, opposite faces come in mutual pairs, so naming one direction names the other.
K.G.B.4Visualize Spatial RelationshipsThis AMC 8 problem only needs Kindergarten cube sense and "6 minus 5 leaves 1" counting you already know!
- Recall the cube's face setup
- Read the three views
- Collect Red's neighbors
- Eliminate to find the opposite
- Flip the relation
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