AMC 10 · 2007 · #11

Grade 3 geometry-3d

Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery

symmetry-argumentdouble-countingmental-arithmetic identify-subproblems ↑ Prerequisites: symmetry-argument

📏 Medium solution 💡 2 insights

Problem

The numbers 1 through 8 are placed on the eight corners of a cube, one number per corner. The arrangement is chosen so that the four corner numbers around every face add up to the same total. Find that common total.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Visualize Spatial Relationships

Trying to actually place 1 through 8 so every face matches is a maze of guessing. The winning move is Tool #17 (Visualize Spatial Relationships): picture the cube and notice that a face and the face directly across from it never share a corner, yet between them they use up all eight corners. That single spatial fact lets Tool #16 (Change Focus) stop worrying about individual placements and instead add up totals, because the two opposite faces together must hold all eight numbers. Tool #1 (Draw a Diagram) keeps the top-face and bottom-face corners straight while doing it.

1STEP 1

Add up all eight numbers

However you arrange them, the corners hold 1 through 8, so the grand total is fixed: 1+2+3+4+5+6+7+8 = 36.

1+2+3+4+5+6+7+8 = 36

2STEP 2

Look at a face and its opposite

The top face uses four corners and the bottom face uses the other four, so the two opposite faces together cover all eight corners once.

4 top corners + 4 bottom corners = 8 vertices (all of them, none repeated)

3STEP 3

Split the grand total in half

Let S be the common face sum. Top plus bottom holds all eight numbers, so S + S = 36 and S = 18, choice (C).

S + S = 36 → 2S = 36 → S = 18 → (C)

Answer

The result 18 is one of the listed choices, and it should land in the middle: the smallest four numbers 1,2,3,4 sum to 10 and the largest four 5,6,7,8 sum to 26, so any face total has to sit between 10 and 26 — and 18 is comfortably inside that range. A different count confirms it: each vertex touches 3 faces, so adding all six face totals counts every number three times, giving 3 × 36 = 108; sharing that across the six faces gives 108 ÷ 6 = 18, the same answer.

💡Key takeaway

Two opposite faces of a cube use all eight corners once, so each face just carries half of the total 1+2+…+8 = 36, which is 18.

Add up all eight numbers
Look at a face and its opposite
Split the grand total in half

Add up all eight numbers

Look at a face and its opposite

Split the grand total in half

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