Problem 13 · 2015 Math Kangaroo
Medium
Algebra & Patterns
sum-constraint
Ella wants to write a number into each circle in the diagram on the right, in such a way that each number is equal to the sum of its two direct neighbours. Which number does Ella need to write into the circle marked with “?”?
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Answer: E — This question has no solution.
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Hint 1 of 2
‘Each number equals the sum of its two neighbours’ rearranges to ‘next = this − previous’, which repeats with period 6.
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Hint 2 of 2
Follow that pattern around the 8-circle ring and see whether the two given numbers, 3 and 5, can both fit.
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Approach: chase the neighbour-sum rule around the ring
- Writing each circle as the sum of its neighbours rearranges to ‘next neighbour = this − previous’, a rule that repeats every 6 steps.
- On a ring of 8 circles this period-6 repetition forces two of the circles (here the ones holding 3 and 5) to carry equal values.
- Since 3 ≠ 5, no consistent filling exists, so the answer is ‘no solution’ (E).
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