Problem 10 · 2014 Math Kangaroo
Medium
Counting & Probability
careful-counting
Gray and white pearls are threaded onto a string (see picture). Tony pulls pearls off from the ends of the string. After pulling off the fifth gray pearl he stops. At most, how many white pearls could he have pulled off?
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Answer: D — 7
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Hint 1 of 2
He can pull from either end, so choose the ends that hand over white pearls cheaply.
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Hint 2 of 2
Count how many white pearls can be removed before the fifth gray pearl is forced off.
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Approach: pull from whichever end yields the most white pearls
- Tony stops the instant the fifth gray pearl comes off, so he removes exactly 5 gray pearls in total, sharing them between the two ends.
- He should pull from whichever end currently hands over white pearls before its next gray pearl.
- Taking the first three gray pearls from one end and the next two from the other end sweeps up the most white pearls along the way.
- Counting the white pearls collected before that fifth gray pearl gives at most 7.
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