I set out to run a 26.5-mile marathon hoping to average less than 9 minutes per mile. As I run, my friends measure my time along various mile-long segments of the course. On each mile that they measure — indeed, on each mile that it’s possible to measure, starting anywhere along the course — my time is exactly 9 minutes. Yet my average for the whole marathon is less than 9 minutes per mile.
How is this possible?
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Solution to Road Work:
I run the first half-mile in 4 minutes and the next in 5 minutes, and continue alternating like this throughout the marathon. In this way I’ll cover any measured mile in 9 minutes. But because the “fast” segments outnumber the “slow” ones, my average pace for the whole run is less than 9 minutes per mile. This works because the course does not cover an even number of miles.
From Frank Morgan, The Math Chat Book, 2000.
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