AMC 10 · 2002 · #18

Grade 6 geometry-3d
Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position
optimizationspatial-visualization extremal-construction ↑ Prerequisites: optimization
📏 Medium solution 💡 2 insights
Problem
A big cube is built from 27 ordinary dice stacked 3 × 3 × 3. On every die the numbers on opposite faces add to 7, and each die may be freely rotated before it is placed. Find the smallest possible total of the numbers showing on the six outer faces of the big cube.

Pick an answer.

(A)
60
(B)
72
(C)
84
(D)
90
(E)
96

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The question asks for a minimum, so Tool #14 (Extreme Principle) drives everything: push every visible face to its smallest legal value. Tool #7 (Identify Subproblems) makes this manageable by splitting the 27 dice into groups that all behave the same way — corner dice show 3 faces, edge dice show 2, face-center dice show 1, and the buried die shows none. Tool #17 (Visualize Spatial Relationships) supplies the key fact that faces meeting at a corner or along an edge are never opposite, so their small numbers can all show at once. Minimize each group, multiply by how many dice are in it, and add.

1STEP 1

Sort the 27 dice by how many faces show

Sort by position: 8 corners show 3 faces, 12 edge dice show 2, 6 face centers show 1, the buried die 0. Check: 8+12+6+1=27.

8 corners(3 faces) + 12 edges(2) + 6 centers(1) + 1 core(0) = 27
2STEP 2

Smallest total on a corner die

Three faces at a corner are never opposite, so they come from three different pairs; take the smaller of each: 1+2+3=6.

min = 1+2+3 = 6
3STEP 3

Smallest totals on edge and face dice

Two faces sharing an edge aren't opposite either, so an edge die drops to 1+2=3 and a face-center die to its single 1.

edge min = 1+2 = 3, face min = 1
4STEP 4

Add up the whole cube

Multiply and add: 8 × 6=48, 12 × 3=36, 6 × 1=6, core 0 — total 90, choice (D).

8(6)+12(3)+6(1)+1(0)=48+36+6=90 → (D)
Answer
90
The total 90 sits right inside the answer range, and a rough estimate agrees: there are 8(3)+12(2)+6(1)=54 visible faces, and the smallest numbers used average a little under 2 per face, so a total near 90 is expected. It must beat the largest choices (96) since we deliberately used the small faces, and it should exceed 60 since corners are forced to include a 3. Only 90 fits both, confirming (D).
💡Key takeaway

Faces that meet at a corner or an edge are never opposite, so a die can show its smallest numbers (1, 2, 3) at once — minimize each die by its position, then add.

  • Sort the 27 dice by how many faces show
  • Smallest total on a corner die
  • Smallest totals on edge and face dice
  • Add up the whole cube

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