Problem 21 · 2019 Math Kangaroo
Stretch
Number Theory
off-by-one
n buttons are placed evenly around a circle. The buttons are labelled clockwise in order with the numbers 1 to n. The button numbered 7 is exactly opposite the button numbered 23. How big is n?
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Answer: B — 32
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Hint 1 of 2
Two buttons are exactly opposite when they are half the circle apart.
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Hint 2 of 2
The gap between button 7 and button 23 is therefore n/2.
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Approach: opposite means a half-circle apart
- If buttons 7 and 23 are exactly opposite, they are n/2 positions apart.
- The gap from 7 to 23 is 23 − 7 = 16, so n/2 = 16.
- Therefore n = 32.
Mark:
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