The Clockwise Ant - Futility Closet

A problem by Argentinian puzzlist Jaime Poniachik, from the February 1992 issue of Games magazine:

An ant crawls onto a clock face at the 6 mark just as the minute hand is passing 12. She begins crawling counterclockwise around the face’s circumference at a uniform speed. When the minute hand passes her, she reverses course and crawls clockwise without changing her speed. Forty-five minutes after her first encounter with the minute hand, it passes her a second time and she departs. How much time did she spend on the clock face?

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54 minutes. Between the ant’s two encounters with the minute hand, the hand passed over 45 minute marks. In that time, the ant passed over 105 minute marks (45 minutes plus one complete circumference). The ratio of their speeds was thus 45/105, or 3/7. If x minutes elapsed before their first encounter, then in that time the minute hand advanced by x minutes while the ant crawled over 30 – x minute marks. So x/(30 – x) = 3/7, which gives x = 9 minutes, and the total time is 9 + 45 = 54 minutes.

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