Competition · AMC preparation · step 4 of 4

AMC 8 · 2024 · #20

Grade 3 geometry-3d
Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position
spatial-visualizationsystematic-enumeration complementary-countingcasework ↑ Prerequisites: spatial-visualizationmental-arithmetic
📏 Medium solution 💡 4 insights 📊 Diagram
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Problem
Pick any three of the eight vertices of cube PQRSTUVW to form a triangle. The question asks: how many of these triangles are equilateral and contain vertex P?

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
3
(E)
6

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Create a Physical Representation

A cube is hard to keep straight in your head, so start with Tool #10 — grab a real cube (a die or a small box), label one corner P, and feel the situation in your hand. The other 7 vertices then naturally sort into just three "distance types" from P. Next, instead of hunting for the vertices that do form an equilateral triangle with P, use Tool #16 (change focus) to rule out the distance types that can't work — that shrinks the candidate set fast. Once the candidates are few, Tool #2 (systematic list) enumerates the triangles, and Tool #3 matches the count to the answer choices.

1STEP 1

Sort the other vertices by distance

Mark P on a real cube and sort the other 7 vertices: 3 edge-neighbors (Q,S,W), 3 across a face (R,T,V), 1 far corner (U) — 3 + 3 + 1 = 7.

3 + 3 + 1 = 7
2STEP 2

Compare edge, face and space diagonals

All six faces are congruent squares, so distances from P split strictly: edge ＜ face diagonal ＜ space diagonal — three different lengths.

edge ＜ face diagonal ＜ space diagonal
3STEP 3

Keep only same-distance vertices

Rule out dead types: space-diagonal has only U; edge length forces a face-diagonal third side. Only the face diagonal survives: {R, T, V}.

candidates = {R, T, V}
4STEP 4

Pick two of the three

Pair P with two of R, T, V: {R,T}→△PRT, {R,V}→△PRV, {T,V}→△PVT — 3 triangles to check.

{R,T}, {R,V}, {T,V} → 3 triangles
5STEP 5

Check each triangle is equilateral

The third side of each is also a face diagonal (RT, RV, TV each span a face), so all sides equal — all three are equilateral.

PR = PT = PV = RT = RV = TV = face diagonal
6STEP 6

Match the count to the choices

(A)0, (B)1, (C)2 are too small (we already exhibited three), and (E)6 exceeds C(3,2)=3 — the count lands on (D).

3 → (D)
Answer
3
For an equilateral triangle with P to exist, at least two more vertices must sit at the same distance type from P. The space-diagonal type has only one member (U), and the edge type fails because P's three edge-neighbors are pairwise a face diagonal apart. So everything has to come from the face-diagonal type — and crucially R, T, V are also a face diagonal apart from each other. Getting exactly 3 triangles also matches the symmetry of the cube: every vertex sits in exactly three such equilateral triangles, which is a nice sanity check.
💡Key takeaway

This AMC 8 problem only needs Grade 3 understanding that shapes in the same category share attributes (every face of a cube is the same square!) you already know!

  • Sort the other vertices by distance
  • Compare edge, face and space diagonals
  • Keep only same-distance vertices
  • Pick two of the three
  • Check each triangle is equilateral
  • Match the count to the choices

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