Problem:
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
Answer Choices:
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Solution:
Working backwards, Fox must have approached the bridge for the third time with coins in order to have no coins left after paying the toll. In the second crossing he must have started with coins in order to have before paying the toll. So he must have started with coins in order to have before paying the toll for the first crossing.
Let be the number of coins Fox had at the beginning. After three crossings he had coins. Setting this equal to 0 and solving gives .
The problems on this page are the property of the MAA's American Mathematics Competitions