Problem 29 · 2013 Math Kangaroo
Stretch
Number Theory
Algebra & Patterns
sum-constraint
Each of the 4 vertices and 6 edges of a tetrahedron is labelled with one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 11 (the number 10 is left out), each used exactly once. The number on each edge is the sum of the numbers on the two vertices it connects. The edge AB has the number 9. With which number is the edge CD labelled?
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Answer: B — 5
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Hint 1 of 2
Add up all ten labels, and note each vertex value is counted in three edges.
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Hint 2 of 2
Find the vertex-sum first; opposite edges (like AB and CD) split that sum.
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Approach: total counts each vertex three times
- The ten labels 1..9 and 11 sum to 56.
- Edges sum to 3×(vertex sum) since each vertex sits on 3 edges, so vertex sum + 3×(vertex sum) = 4×(vertex sum) = 56, giving vertex sum 14.
- Edges AB and CD together use all four vertices, so (A+B) + (C+D) = 14.
- With AB = 9, CD = 14 − 9 = 5.
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