Solution to Riddle of the Week: The Trains and Grandmas Problem

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Kory Kennedy using Illustration Copyright

This is a solution to Two Trains. Two Grandmas. Can You Solve This Tricky Math Riddle?, part of our Riddle of the Week series.


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Jesse’s two grandmothers want to see him every weekend, but they live on opposite sides of town. As a compromise, he tells them that every Sunday, he’ll head to the subway station nearest to his apartment at a random time of the day and will hop on the next train that arrives.

If it happens to be the train traveling north, he’ll visit his Grandma Erica uptown, and if it happens to be the train traveling south, he’ll visit his Grandma Cara downtown. Both of his grandmothers are okay with this plan, since they know both the northbound and southbound trains run every 20 minutes.

But after a few months of doing this, Grandma Cara complains that she sees him only one out of five Sundays. Jesse promises he’s indeed heading to the station at a random time each day. How can this be?


Although both trains come exactly every 20 minutes, the timing still matters: Suppose the northbound train comes on the hour, on the 20, and on the 40, so at 9:00, 9:20, 9:40, 10:00, etc., but the southbound train comes on the 4, 24, and 44, so 9:04, 9:24, 9:44, 10:04, etc.

This means in any hour, there will be only 12 minutes in which the southbound train will be the next train to arrive. That is, if Jesse arrives between 9:00-9:04, between 9:20-9:24, or between 9:40-9:44, he’ll get on the southbound train. Otherwise, he’ll get on the northbound train.

Since Jesse arrives at a random point every day (or equivalently at a random point in each hour), his chance of getting on the southbound train will be 12 out of 60, or one-fifth. And so that’s the answer: Both the northbound and the southbound trains arrive at the station every 20 minutes, but the southbound train always arrives 4 minutes after the northbound train.

Grandma Cara may be lucky, though. If the southbound train was scheduled to arrive only 1 minute after the northbound train, Jesse would only visit two to three times per year!

Want to talk about brain teasers or try to out-riddle the riddler? Find Laura on Twitter at @LauraFeiveson.

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