Problem 27 · 2025 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintsubstitution
A number is to be written in each circle of the diagram in such a way that the sum of the numbers in three touching circles is always the same. Some of the numbers are already given. What is the sum of all the numbers in the middle row?
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Answer: C — 13
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Hint 1 of 3
Every set of three mutually touching circles forms a tiny triangle with the same sum; use both the upward- and downward-pointing triangles.
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Hint 2 of 3
Two triangles that share the same pair of circles must have equal third circles, which lets you copy values between rows.
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Hint 3 of 3
Propagate from the given 4 (top), 2 (right of middle) and 1 (bottom) until every middle circle is fixed.
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Approach: use the equal-triple-sum constraints to propagate values
- The circles pack in three rows (4, 5, 4); each small triangle of three touching circles has the same sum \(S\), and two triangles sharing a side force their opposite circles to be equal.
- Chaining these equalities from the known 4, 2 and 1 fixes \(S=7\) and fills the middle row as \(4,\,2,\,1,\,4,\,2\).
- Their sum is \(4+2+1+4+2=\) 13, answer C.
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