AMC 10 · 2003 · #10
Grade 6 geometry-3dThe polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A flat shape made of $4$ congruent squares joined edge to edge is shown. A fifth congruent square can be attached to the border at any one of $9$ marked positions, one at a time, giving $9$ different flat shapes. For how many of these $9$ shapes can the paper be folded along the square edges into a cube that is missing exactly one face?
Givens: The base shape is $4$ congruent squares joined edge to edge (a flat strip of squares).; A $5$th congruent square is attached at exactly one of the $9$ indicated edge positions, one position per trial.; Folding is allowed only along the shared edges between squares.; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$.
Unknowns: How many of the $9$ resulting $5$-square shapes fold into a cube with one face missing.
Understand
Restated: A flat shape made of $4$ congruent squares joined edge to edge is shown. A fifth congruent square can be attached to the border at any one of $9$ marked positions, one at a time, giving $9$ different flat shapes. For how many of these $9$ shapes can the paper be folded along the square edges into a cube that is missing exactly one face?
Givens: The base shape is $4$ congruent squares joined edge to edge (a flat strip of squares).; A $5$th congruent square is attached at exactly one of the $9$ indicated edge positions, one position per trial.; Folding is allowed only along the shared edges between squares.; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$.
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #1 Draw a Diagram, #16 Change Focus / Count the Complement
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 $9$.
Execute — Answer: E
6.G.A.4 Step 1 Restate the goal as 5 faces
- A cube has $6$ faces.
- "Missing one face" means $6-1=5$ faces are present, so the target is an open box: $5$ squares that fold onto $5$ different faces of a cube.
- Every candidate here has $4$ base squares plus $1$ added square, which is exactly $5$ squares.
- So the shape has the right number of pieces; the only question is whether all $5$ can settle onto distinct faces when folded.
💡 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.4 Step 2 The 4 base squares already fold cleanly
- Fold the original $4$-square shape up.
- Because the squares form a connected strip bent around the corner of a cube, they wrap onto $4$ different faces without any two overlapping — they build $4$ of the $5$ faces of the open box, leaving $2$ face slots still empty.
- So the base is already a good partial box, and adding a fifth non-overlapping square anywhere just fills one more face and completes a cube-minus-one-face.
💡 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.4 Step 3 A position works unless the new square overlaps
- Since the $4$ base faces are fixed, the only way an added square can fail is if, when folded, it swings onto a face that is already occupied by one of the base squares — two squares on the same face means it is not a valid net.
- So the rule becomes simple: an attachment position works exactly when the folded fifth square lands on an empty face, and fails exactly when it overlaps a face that is already used.
💡 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.
6.G.A.4 Step 4 Count the failures and subtract
- Mentally fold the added square at each of the $9$ marked spots.
- At $3$ of the positions the square folds back onto a face already taken by a base square, so those overlap and fail.
- At the remaining $6$ positions the square swings onto an empty face and the shape folds into a cube with one face missing.
- Counting the complement: $9 - 3 = 6$ positions work, which is 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.4 A cube has $6$ faces. "Missing one face" means $6-1=5$ faces are present, so the 6.G.A.4 Fold the original $4$-square shape up. Because the squares form a connected stri 6.G.A.4 Since the $4$ base faces are fixed, the only way an added square can fail is if, 6.G.A.4 Mentally fold the added square at each of the $9$ marked spots. At $3$ of the po Review
Reasonableness: The base $4$ squares already form a working open box, and most of the $9$ marked edges point outward into empty space, so we should expect many more successes than failures — a large value like $6$ is believable while a tiny value like $2$ would be surprising. The only positions that can fail are the ones where the added square folds directly onto a face the base already uses, and exactly $3$ of the marked spots do that. So $9-3 = 6$ work, matching (E) and not the smaller choices.
Alternative: Build it physically (Tool #10, Create a Physical Representation): cut the $4$-square base from paper and fold it into an open box, then hold a fifth square against each of the $9$ edges and try to fold it flat onto a face. The $3$ squares that would have to cover an already-standing wall are the failures; the other $6$ drop onto an open face, again giving $6$.
CCSS standards used (min grade 6)
6.G.A.4Represent three-dimensional figures using nets and find surface area (Folding the flat squares into a cube-with-one-face-missing and checking, at each of the 9 positions, whether the added square lands on an empty face or overlaps a face already used.)
⭐ A 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 $6$.
⭐ A 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 $6$.
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