AMC 10 · 2016 · #15
Grade 6 geometry-3dPick an answer.
The question asks for the greatest possible value, so this is an optimization — tool #14 (Extreme Principle) is the spine. But before we can push the sum to its maximum, we need a clean formula for it. Tool #17 (Visualize Spatial Relationships) shows that each vertex picks one face from each pair of opposite faces, and tool #4 (Introduce a Variable) lets us name the opposite pairs and factor the messy eight-term sum into the tidy product (a+a')(b+b')(c+c'). Once the sum is a product of three pieces whose total is fixed, the Extreme Principle tells us to make the pieces equal, and tool #6 (Guess and Check) confirms an equal split is actually buildable and that lopsided splits score lower.
See what meets at a vertex
Each corner takes one face from each pair.
Opposite faces can't touch the same corner, so each corner is forced to take one number out of each opposite pair.
6.G.A.4Visualize Spatial RelationshipsName the opposite pairs
Naming the pairs makes the structure visible.
Giving the opposite faces letters turns 'eight corners of a cube' into eight neat algebra terms.
6.EE.A.2Introduce A VariableFactor the eight-term sum
The whole sum factors into three brackets.
Multiplying out (a+a')(b+b')(c+c') produces exactly one term per corner, so the sum and the product are the same thing.
Multiplying the three pair sums produces exactly one term per corner, so the sum and the product are the same thing.
▸ Why?
Expanding a product of sums spreads every choice across the others, giving one term per combination.
▸ Why?
Each corner picks one face from each opposite pair, and those picks are made independently.
Make the three sums equal
Their total is fixed, so make them equal.
With a fixed total, balancing the parts beats lopsiding them — spreading the sum evenly packs in the most product.
6.EE.A.1Extreme PrincipleBuild it and check
A real arrangement gives 729, choice (D).
The perfectly balanced pairing (2,7),(3,6),(4,5) exists, so the ideal 9×9×9 isn't just a dream — it's reachable.
6.EE.A.1Guess And CheckThe cube-sum is just (top+bottom)(front+back)(left+right), and three numbers with a fixed total multiply to the most when you split them evenly — so make every opposite pair add to 9.
- See what meets at a vertex
- Name the opposite pairs
- Factor the eight-term sum
- Make the three sums equal
- Build it and check