<p>Consider a cube, if all its six faces are assigned with a unique number from 2, 3, 4, 5, 6, 7 with one number at each face. For each of the eight vertices of the cube, a product of three numbers where the three numbers are the numbers assigned to the three faces that include the vertex. What is the largest value of the sum of these eight products?</p>
Step-by-Step Solution
Key Concept: Each of the 8 vertex products equals the product of 3 face numbers meeting at that vertex. The sum of all 8 products equals (sum of products of opposite face pairs) times (sum of all numbers) divided strategically, but more directly: if opposite faces have values a,a' and b,b' and c,c', then the sum = (a+a')(b+b')(c+c'). To maximize, pair numbers to maximize this expansion.
<p><strong>Step 1:</strong> Understand vertex structure. Each vertex is the intersection of exactly 3 faces. If opposite faces are labeled {a, a'}, {b, b'}, {c, c'}, then the 8 vertices have products: ab c, ab c', a b'c, ab'c', a'bc, a'bc', a'b'c, a'b'c'.</p><p><strong>Step 2:</strong> Sum all 8 products: (a+a')(b+b')(c+c'). This is the complete expansion of the product of three binomials.</p><p><strong>Step 3:</strong> We must partition {2,3,4,5,6,7} into 3 pairs of opposite faces. We need to choose which numbers are a,a'; b,b'; c,c'.</p><p><strong>Step 4:</strong> Expand: (a+a')(b+b')(c+c') = abc + abc' + ab'c + ab'c' + a'bc + a'bc' + a'b'c + a'b'c'. To maximize this sum, pair opposite faces as: {2,7}, {3,6}, {4,5}.</p><p><strong>Step 5:</strong> Calculate: (2+7)(3+6)(4+5) = 9 × 9 × 9 = 729.</p><p>∴ Answer: <strong>729</strong></p>
Correct Answer: 729