AMC 10 · 2013 · #14
Easy mode Grade 3Start with a solid cube whose sides are 3 long. At each of its 8 corners, cut off a small solid cube whose sides are 1 long. How many edges does the leftover solid have?
Pick an answer.
AMC 10 2013 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 solid cube of side length $3$ has a solid unit cube (side length $1$) sliced off at each of its $8$ corners. Count how many edges the leftover solid has.
Givens: The starting shape is a solid cube of side length $3$; A solid unit cube of side length $1$ is removed from each of the $8$ corners; A cube has $12$ edges, $6$ faces, and $8$ vertices; Answer choices: (A) $36$, (B) $60$, (C) $72$, (D) $84$, (E) $108$
Unknowns: The number of edges on the remaining solid after all $8$ corner cubes are removed
Understand
Restated: A solid cube of side length $3$ has a solid unit cube (side length $1$) sliced off at each of its $8$ corners. Count how many edges the leftover solid has.
Givens: The starting shape is a solid cube of side length $3$; A solid unit cube of side length $1$ is removed from each of the $8$ corners; A cube has $12$ edges, $6$ faces, and $8$ vertices; Answer choices: (A) $36$, (B) $60$, (C) $72$, (D) $84$, (E) $108$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #17 Visualize Spatial Relationships, #1 Draw a Diagram
Counting every edge at once is confusing, so Tool #7 (Identify Subproblems) splits the total into two independent counts: how many of the original cube's edges survive, and how many brand-new edges the $8$ identical corner cuts add. Tool #17 (Visualize Spatial Relationships) pictures a single corner notch to see that the three freshly exposed square faces contribute exactly $9$ edges. Tool #1 (Draw a Diagram) sketches that one corner so the edges shared between the new faces are not double-counted. Because all $8$ corners are identical, one careful local count multiplies up to the whole solid.
Execute — Answer: D
2.G.A.1 Step 1 Keep the original cube's edges
- First count the edges that were already there.
- A unit cube is sliced off each end of every edge of the big cube.
- Slicing a unit piece off the two ends of an edge only shortens that edge; it does not split it into pieces or erase it.
- A cube has $12$ edges, so all $12$ original edges are still present.
💡 Trimming both tips of a stick leaves one shorter stick, not zero and not two.
K.G.B.4 Step 2 See what one corner cut exposes
- Now look at a single corner.
- The removed unit cube had $6$ faces: $3$ of them sat on the outside surface of the big cube, and the other $3$ were buried inside it.
- Removing the little cube uncovers those $3$ inside faces, so each corner cut opens up exactly $3$ new unit-square faces, and these three squares all meet at one inner point.
💡 Scooping a cubic bite out of a corner reveals the three inside walls of the bite.
2.G.A.1 Step 3 Count the new edges at that corner
- Each of the $3$ exposed squares has $4$ edges, giving $3 \times 4 = 12$ edge-slots.
- But the three squares meet at the inner point, and every pair of them shares one edge running to that point, so $3$ edges are each counted twice.
- The number of distinct new edges at one corner is $12 - 3 = 9$.
💡 Where two faces meet they share an edge, so shared edges must be counted once, not twice.
3.OA.D.8 Step 4 Add up over all eight corners
- All $8$ corners are identical, so each adds $9$ new edges: $8 \times 9 = 72$ new edges.
- Combine these with the $12$ surviving original edges to get $12 + 72 = 84$.
- The remaining solid has $84$ edges, which is choice (D).
💡 Eight identical notches mean you count one notch and multiply, then add the untouched edges.
2.G.A.1 First count the edges that were already there. A unit cube is sliced off each en K.G.B.4 Now look at a single corner. The removed unit cube had $6$ faces: $3$ of them sa 2.G.A.1 Each of the $3$ exposed squares has $4$ edges, giving $3 \times 4 = 12$ edge-slo 3.OA.D.8 All $8$ corners are identical, so each adds $9$ new edges: $8 \times 9 = 72$ new Review
Reasonableness: Check the total with Euler's polyhedron formula $V - E + F = 2$. Faces: the $6$ original faces survive (each big face just loses its four corner squares but stays one connected face), plus $3$ new faces per corner give $3 \times 8 = 24$, so $F = 6 + 24 = 30$. Vertices: each corner cut deletes the $1$ original corner point but creates the little cube's other $7$ corners, so $V = 8 \times 7 = 56$. Then $E = V + F - 2 = 56 + 30 - 2 = 84$, matching the direct count and confirming (D). The value $84$ is also one of the listed choices and sensibly larger than the original $12$ edges.
Alternative: Instead of separating old and new edges, count everything through Euler's formula from the start: determine that the carved solid has $V = 56$ vertices and $F = 30$ faces, then solve $V - E + F = 2$ for $E = V + F - 2 = 56 + 30 - 2 = 84$. This swaps the edge visualization for a face-and-vertex census and lands on the same answer (D).
CCSS standards used (min grade 3)
2.G.A.1Recognize and draw shapes having specified attributes (Recalling that a cube has $12$ edges and that each newly exposed face is a unit square with $4$ edges.)K.G.B.4Analyze and compare two- and three-dimensional shapes (Seeing that removing one corner unit cube uncovers exactly $3$ inside square faces that meet at one point.)3.OA.D.8Solve two-step word problems using four operations within 100 (Combining the two counts as $12 + 8 \times 9 = 84$.)
⭐ Cutting a cube off a corner does not erase the old edges; it just adds a little three-sided notch worth $9$ new edges, and eight notches pile on $72$ more.
⭐ Cutting a cube off a corner does not erase the old edges; it just adds a little three-sided notch worth $9$ new edges, and eight notches pile on $72$ more.
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