AMC 10 · 2016 · #14

Grade 7 geometry-3d
Archetype Casework by Position / Vertex Type └ Sub-archetype Vertex or Element Role
spatial-visualizationcombinations-basic caseworksymmetry-argument ↑ Prerequisites: spatial-visualization
📏 Long solution 💡 4 insights
Problem
Eight numbers label a cube's corners so every face totals the same. Count the labelings up to rotation.

Pick an answer.

(A)
1
(B)
3
(C)
6
(D)
12
(E)
24
How to solve
Strategy Make a Systematic List

The question asks "how many," so tool #2 (Make a Systematic List) drives the count once the structure is pinned down. First tool #17 (Visualize Spatial Relationships) fixes the cube facts — each corner on three faces, each edge on two — which forces the common face total and reveals where the extreme numbers 1 and 8 must sit. Tool #3 (Eliminate Possibilities) then rules out every placement except 1 and 8 sharing an edge, collapsing a messy search into three clean cases. Tool #1 (Draw a Diagram) keeps the two faces meeting at that edge, and the edge across the cube, straight in mind while the pairs are placed.

1STEP 1

Every face totals 18

Every face totals 18.

1+2+…+8=36, 6S=3 × 36=108, S=18
2STEP 2

1 and 8 must share an edge

The extremes are forced onto one edge.

N=o+2 ≤ 9, N'=o'+16 ≥ 18, {2,3,4}∩{5,6,7}=∅
3STEP 3

Fill each face with a pair summing to 9

The rest pair up to a fixed sum.

{2,7}, {3,6}, {4,5}; far-edge pair: 3 choices
4STEP 4

Each case gives a shape and its mirror

Three cases, each with a mirror, give 6, choice (C).

3 cases × 2 mirrors=6 → (C)
Answer
6
The face total 18 is forced, and the count 6 sits in the middle of the answer list, fitting that rotations (but not reflections) are merged. Each of the three pairs {2,7},{3,6},{4,5} takes one turn on the far edge, and the two mirror copies in each case are genuinely different labelings, so 3 × 2=6 with nothing double-counted and nothing missed.
💡Key takeaway

Every face adds to 18, that forces 1 and 8 onto one edge, and then you just pick which pair sits on the far edge (3 ways) and a mirror flip (2 ways) — six arrangements in all.

  • Every face totals 18
  • 1 and 8 must share an edge
  • Fill each face with a pair summing to 9
  • Each case gives a shape and its mirror

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