Problem 25 · 2018 Math Kangaroo
Stretch
Algebra & Patterns
sequence-of-figures
A number is written at every vertex of the 18-sided shape so that each number equals the sum of the numbers at its two neighbouring vertices. Two of the numbers are given (see picture). Which number is written at vertex A?
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Answer: D — 38
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Hint 1 of 2
The rule “each = sum of its two neighbours” makes the sequence of vertex values repeat with period 6.
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Hint 2 of 2
Write a few terms: the pattern is \(x,\ y,\ y-x,\ -x,\ -y,\ x-y\) and then it repeats.
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Approach: use the period-6 pattern forced by the neighbour rule
- If a vertex equals the sum of its neighbours, going around gives the repeating block \(x,\,y,\,y-x,\,-x,\,-y,\,x-y\) (period 6).
- Over the 18 vertices this block repeats exactly three times.
- Placing the two given values (20 and 18) at their vertices and following the period-6 pattern forces vertex A to 38.
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