AMC 10 · 2013 · #14
Grade 3 geometry-3dPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Keep the original cube's edges
Slicing a unit cube off each end only shortens an edge, never splits or erases it, so all 12 original edges survive.
Trimming both tips of a stick leaves one shorter stick, not zero and not two.
2.G.A.1Visualize Spatial RelationshipsSee what one corner cut exposes
The little cube's buried faces get uncovered, so each corner opens 3 new unit squares that all meet at one inner point.
Scooping a cubic bite out of a corner reveals the three inside walls of the bite.
K.G.B.4Visualize Spatial RelationshipsCount the new edges at that corner
The 3 squares give 3 × 4 = 12 edge-slots, but each pair shares one edge, so 12 - 3 = 9 new edges are distinct.
Where two faces meet they share an edge, so shared edges must be counted once, not twice.
Where two new faces meet they share an edge, so shared edges must be counted once, not twice.
▸ Why?
Adding each face's edges counts the shared ones more than once, so the overlaps must come out.
▸ Why?
A solid's corners, edges, and faces are tied together, so a correct edge count has to be consistent.
Add up over all eight corners
The 8 identical corners add 8 × 9 = 72 edges; with the 12 survivors that is 12 + 72 = 84, choice (D).
Eight identical notches mean you count one notch and multiply, then add the untouched edges.
3.OA.D.8Identify SubproblemsCutting 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.
- Keep the original cube's edges
- See what one corner cut exposes
- Count the new edges at that corner
- Add up over all eight corners