AMC 10 · 2009 · #20

Grade 7 geometry-3dcounting

Archetype Casework by Position / Vertex Type └ Sub-archetype Count of Type

spatial-visualizationdouble-countingface-adjacency identify-subproblemsphysical-representation ↑ Prerequisites: spatial-visualization

📏 Long solution 💡 3 insights

Problem

Every corner of a convex solid is sliced off by a plane cutting only that corner's edges, with no two planes meeting. Count the edges of the solid that remains.

Pick an answer.

(A)

200

(B)

(C)

300

(D)

400

(E)

How to solve

Strategy Visualize Spatial Relationships

Nothing here can be computed until it is clear what one single cut does, so the first move is to picture one corner up close and see exactly which pieces appear. Once one cut is understood, the whole count is local: every edge of R is either the surviving middle of an old edge or a side of one of the new flat faces. Adding the new sides corner by corner is the same as counting the two ends of every old edge, which is a second way of organizing the same total and is where the mysterious n cancels out. Finally, since two of the answer choices still contain n, a deliberate check that the total cannot depend on n settles the choice.

1STEP 1

See one cut up close

One cut crosses exactly the edges at that corner.

P_k crosses exactly the d_k edges at V_k

2STEP 2

Every edge is cut twice

So every old edge is cut exactly twice.

100 edges × 2 cuts each = 200 cut points

3STEP 3

Each cut leaves a polygon face

Each cut leaves a new polygon face.

new edges from P_k = d_k

4STEP 4

Count the new edges

Those faces contribute 200 new edges in total.

Σ_k=1ⁿ d_k = 2 × 100 = 200

5STEP 5

Check nothing is missed

The middle stretches of the old edges survive.

edges of R = (middle segments) + (sides of the n new faces)

6STEP 6

Add up and rule out n

Adding gives 300, with the corner count never entering.

100 + 200 = 300

7STEP 7

Sanity-check with a cube

A cube checks the formula, confirming 300, choice (C).

E_R = 3E_Q, 3 × 12 = 36, 3 × 100 = 300

Answer

300

A second local count agrees. Each new corner of R is a cut point on an old edge, and exactly three edges of R meet there: the middle segment of that old edge, and the two sides of the new polygon that end at that point. With 200 new corners each of degree 3, and each edge having two ends, the edge count is (3 × 200)/2 = 300, matching. The number is also plausible in size: R has all 100 old edges still partly present plus one new ring of edges around each corner, so a total a few times larger than 100 is expected, while 200 would be too few to include both and 400 would require new edges that no face of R could hold.

💡Key takeaway

Slicing off every corner keeps the middle of each edge and turns each of its two ends into a new edge, so the solid ends up with three times as many edges as it started with.

See one cut up close
Every edge is cut twice
Each cut leaves a polygon face
Count the new edges
Check nothing is missed
Add up and rule out n
Sanity-check with a cube