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March 26, 2008
Follow the Bandits
Philo, a Greek merchant, is traveling across a vast desert. Suddenly, he is attacked by a group of bandits who steal his bundles of silk and ride off into the wilderness. Badly shaken, Philo sits down in the meager shade of a lone, scraggly palm tree to plan what to do next. He decides to search for the thieves, following their tracks.
A few weeks later, Philo arrives at a city and spies the bandits, who promptly flee into the desert. Philo attempts to follow them, but he quickly loses sight of the culprits. However, he knows from overhearing their plans that they will visit seven other cities to sell the silk they have stolen and then return to the city where he is now.
In order to recover all his silk, Philo wants to visit the cities in the same order as they do, but he does not know what that order will be. How many different routes could the bandits take?
—Emma Pierson, Grade 6
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How many different routes could the bandits take if they visited one other city? two other cities? three other cities? four other cities? Do you see a pattern?
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On the first leg, the bandits could go to any one of seven cities, so there are seven possible routes. They have six cities left for the next stage, then five, four, and so on. Altogether, they could follow any one of 7 × 6 × 5 × 4 × 3 × 2 × 1, or 5,040, different routes before returning to their starting point. So, by randomly picking a route, Philo has a 1 in 5,040 (or 0.019 percent) chance of finding the right one.
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