Problem 17 · 2020 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraint
Matias wrote 15 numbers around the wheel shown. Only one of them is visible, the 10 at the top. The sum of the numbers in any seven consecutive positions (such as the gray positions in the figure) is always the same. When seven numbers in consecutive positions are added, which of the following results is possible?
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Answer: B — 70
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Hint 1 of 2
Equal 7-sums force entry n to equal entry n−7 around the 15-position wheel.
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Hint 2 of 2
Since gcd(7,15)=1, that links all the numbers.
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Approach: constant window sum forces all entries equal
- If every 7 consecutive entries have the same sum, then sliding by one shows entry n = entry n−7 (indices mod 15).
- Because gcd(7, 15) = 1, this makes all 15 numbers equal; the visible one is 10, so all are 10.
- Then any 7-sum is 7×10 = 70.
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