Problem 12 · AMC 8 Stretch
Stretch
Number Theory
Geometry & Measurement
pattern-recognitionlogical-reasoning
Mark a point on a circle. Spin it by a fixed angle to get the next dot, spin again, and so on. If you spin by \(40^\circ\) each time, how many dots are there before you land back exactly on the starting point?
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Answer: 9 dots
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Hint 1 of 4
You are back at the start when the total amount you've spun adds up to a whole number of full circles.
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Hint 2 of 4
A full circle is \(360^\circ\). Keep adding \(40^\circ\): 40, 80, 120, ...
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Hint 3 of 4
When do you first reach a multiple of 360?
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Approach: Find when the total spin first completes whole circles
- You return to the start exactly when your total spin is a whole number of complete circles (\(360^\circ, 720^\circ, \dots\)).
- Spinning \(40^\circ\) each time, after \(n\) spins the total is \(40n\) degrees. The first time this is a multiple of 360 is when \(40n = 360\), so \(n = 9\).
- So there are 9 dots before returning. (For comparison, spinning \(90^\circ\) would give 4 dots, a square.)
- The answer is 9 dots.
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