AMC 10 · 2003 · #10
Grade 6 geometry-3d
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a folding question, so the core skill is Tool #17 (Visualize Spatial Relationships): mentally fold the flat squares up around a cube and watch where each square lands. Tool #1 (Draw a Diagram) keeps the count honest — sketch the base shape and mark the 9 spots so none is missed. The clever move is Tool #16 (Change Focus / Count the Complement): instead of hunting for the positions that work, notice the base 4 squares already fold into 4 good faces, so a new square works unless it collides with a face that is already taken. Counting the few collisions is far easier than checking all 9 successes, so count the failures and subtract from 9.
Restate the goal as 5 faces
A cube has 6 faces, so missing one leaves 5 — an open box — and 4 base squares plus 1 added square is exactly 5.
A cube with a lid removed is just five squares wrapped around, so you are building an open box out of five faces.
6.G.A.4Visualize Spatial RelationshipsThe 4 base squares already fold cleanly
Folded up, the 4-square strip wraps 4 different faces with no overlap, so four of the five walls already stand.
Four of the five walls are already standing, so a fifth square only needs to land on empty air, not on top of a wall that is already there.
6.G.A.4Visualize Spatial RelationshipsA position works unless the new square overlaps
So an added square fails only if it folds onto a face a base square already occupies; land on empty air and the position works.
Overlap is the only enemy: if the new square does not double up on an existing face, five distinct faces automatically make a cube missing one face.
Overlap is the only enemy: if the new square does not double up on a face, five distinct faces fold into an open box.
▸ Why?
The open box is exactly five faces put together, so five distinct faces with no gaps is all it takes.
▸ Why?
Folding moves each square without stretching it, so a square that lands on empty air really becomes a face.
Count the failures and subtract
Fold each of the 9 marked spots mentally: 3 overlap a used face and fail, so 9 - 3 = 6 positions work — choice (E).
Only a handful of spots fold back onto an occupied face, so counting those 3 bad ones and subtracting is quicker than checking all nine.
6.G.A.4Change Focus Count The ComplementA cube missing one face is just five squares folded into an open box; the four given squares already make four walls, so a new square works unless it folds on top of a wall that is already there — and only 3 of the 9 spots do that, leaving 6.
- Restate the goal as 5 faces
- The 4 base squares already fold cleanly
- A position works unless the new square overlaps
- Count the failures and subtract
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