AMC 8 · 1999 · #8

Grade 6 geometry-3d

Archetype Net Folding / Face Adjacency └ Sub-archetype Fold Net to Locate Opposite or Adjacent Face

spatial-visualizationpolyhedron-netsface-adjacencynet-folding physical-representationcasework ↑ Prerequisites: spatial-visualization

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Six squares — colored Red, Blue, Orange, Yellow, Green, White (each color on both sides) — are hinged together in the net shown and then folded into a cube. Which color ends up on the face opposite White?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Visualize Spatial Relationships

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.

1STEP 1

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.

base = Y → G, O, W each share a hinge with Y

2STEP 2

Fold the three neighbors up: Green becomes the left face, Orange the right face, and White the bottom face.

Front=Y, Left=G, Right=O, Bottom=W

3STEP 3

The strip hinged above Green folds next: Blue swings onto the top face, and Red continues onto the back face.

Top=B, Back=R

4STEP 4

White sits on the bottom, so the face directly opposite it is the top — Blue.

Bottom=W → opposite face=Top=B → (A)

Answer

Adjacency sanity check: W is hinged to Y in the net, so W and Y are adjacent (cannot be opposite). After folding, W is the bottom face and touches every side face except the top, so W is adjacent to Y (front), G (left), O (right), and R (back). The only face W does not touch is the top face, which is B — confirming (A). A second check uses a known cube-net rule: in a straight strip of three squares in a net, the two outer squares fold to opposite faces of the cube. The strip W-Y-(square above Y) in the net would put W opposite that top square. The square above Y in the original net is empty, but the chain W → Y → G → B shows that B rises to the top and W stays at the bottom, again giving W opposite B.

💡Key takeaway

Pin 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.

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