AMC 10 · 2016 · #17
Grade 6 geometry-3dPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
The three faces at any corner are never opposites — a vertex takes one number from each of the three opposite pairs.
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
Label the opposite pairs (a,a'), (b,b'), (c,c'); the eight vertex products each pick one letter from every pair.
Giving the opposite faces letters turns 'eight corners of a cube' into eight neat algebra terms.
6.EE.A.2Use Matrix LogicFactor the eight-term sum
Group and factor: the eight products collapse into the single product (a+a')(b+b')(c+c') of the three pair-sums.
Multiplying out (a+a')(b+b')(c+c') produces exactly one term per corner, so the sum and the product are the same thing.
6.EE.A.3Use Matrix LogicMake the three sums equal
The three pair-sums always total 27, and a fixed sum multiplies to the most when split evenly — here 9 × 9 × 9.
With a fixed total, balancing the parts beats lopsiding them — spreading the sum evenly packs in the most product.
6.EE.A.1Evaluate Finite DifferencesBuild it and check
The pairing (2,7), (3,6), (4,5) makes every pair sum to 9, so the total reaches 9 · 9 · 9 = 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