Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #8
Grade 6 geometry-3d
Pick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem hands us a 2D net and asks about a 3D folded cube — the canonical trigger for Tool #17 (Visualize Spatial Relationships). Pick one square as the base, then mentally fold the rest around it and track where each color lands. Tool #10 (Create a Physical Representation) is the safety net: anyone unsure of the mental fold can copy the net onto paper, label the colors, and physically fold it — touching the problem is encouraged for 3D geometry. Tool #3 (Eliminate Possibilities) lets us double-check by noticing that the three faces hinged directly to W in the net (Y, plus the two faces reached through Y) must be adjacent to W, so they cannot be opposite to it — that crosses three answer choices off the list.
Pick a base face
Pin Yellow as the front face. Green, Orange, and White each share a hinge with Yellow, so all three fold up into sides touching it.
Grade 6: a cube net is the unfolded surface of a cube. Whatever square shares a hinge with the base becomes one of the four faces touching the base.
6.G.A.4Visualize Spatial RelationshipsFold the three side faces
Fold the three neighbors up: Green becomes the left face, Orange the right face, and White the bottom face.
If you cut the net out of paper, this is what your hands do first: lift each neighbor of Y until it stands up perpendicular to Y.
6.G.A.4Create A Physical RepresentationFold the last two faces
The strip hinged above Green folds next: Blue swings onto the top face, and Red continues onto the back face.
Chains of hinged squares fold like a hinge of a folding ruler: each new square swings around the next available outer edge of the partly-built cube.
When the hinged net folds into the cube, the Blue square lands on the top face while the White square folds down to the bottom, so Blue ends up directly opposite White.
▸ Why?
Folding each paper hinge is a rigid flip, so every square keeps its shape and each hinge turns into a shared cube edge; tracing the squares through the fold fixes exactly where White and Blue land.
▸ Why?
White is hinged straight below Yellow, so one right-angle fold swings it from beside Yellow to underneath — the bottom face — and a rigid fold makes that quarter turn exact.
▸ Why?
Blue is carried up by Green: after Green folds up into the left wall, Blue (hinged to Green's far edge) folds once more over the top edge and lies flat on the top face.
▸ Why?
On a cube the top and bottom faces never share an edge, so they are one of the three opposite pairs; a square resting on the bottom is therefore opposite the square on the top.
▸ Why?
The cube's six faces split with no gaps or overlaps into three pairs that share no edge — top with bottom, front with back, left with right — so each face is opposite exactly one other.
Find the face opposite W
White sits on the bottom, so the face directly opposite it is the top — Blue.
Cross-check with the answer choices: Y, G, O all sit next to W in the folded cube, so they cannot be opposite to W — that eliminates (E), (B), (C). And R shares the back-top edge with the top face B, so R is adjacent to B, not equal to B — that eliminates (D). Only (A) remains.
6.G.A.4Eliminate PossibilitiesPin one square as the base, fold the rest around it, and read off the opposites — Grade 6 net-folding tells us the face across from W is B.
- Pick a base face
- Fold the three side faces
- Fold the last two faces
- Find the face opposite W
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