The Commuter's Choice

This puzzle was suggested by Martin Gardner in his book My Best Mathematical and Logic Puzzles.

Mr. Tsuru commutes each day by train between his home and the downtown office building where he works. Trains run back and forth on the line from the grey station at one end to the pink station at the other end. There are dozens of stations. We’ve shown only five of them. Mr. Tsuru works near the blue station. His home is halfway between the red and yellow stations, so he can use either to get home in the same time.

communters_choice Trains constantly come through the blue station in both directions. So, when he leaves work, Mr. Tsuru can take the train in either direction. His approach is to take the train that arrives first regardless of the direction it is traveling. If it is going one way, he will end up at the red station. If it is going the other way, he will arrive at the yellow station.

In each direction, the trains are spaced exactly ten minutes apart. Mr. Tsuru works very irregular hours, so the time he leaves work varies greatly and the time at which he reaches the blue station is essentially random.

He expects that he will be going home via the red station about half the time and via the yellow station the other half. But, after several months of commuting, he finds that he is using the red station about 90% of the time. How is this possible?

Course:

Math [1]

Result/Solution(s)

Solution: The Commuter's Choice Math Puzzle

We said that the trains in each direction were spaced at ten-minute intervals, but we did not say how those intervals overlapped with each other. One possibility would be for the trains from each direction to arrive simultaneously. Then ten minutes later, two more trains would arrive simultaneously, one from each direction. In this case, Mr. Tsuru would have a hard time choosing.

Another possibility is that every five minutes a train would arrive, first from one direction, then from the other, again from the original direction, and so on. With this spacing, Mr. Tsuru, arriving at a random time at the station, would be equally likely to go in one direction or the other.

But what if a train arrives heading toward the red station, followed one minute later by a train heading toward the yellow station? Then nine minutes later, another train arrives headed toward the red station. This is ten minutes after the first red-bound train. Then a minute later a train headed toward the yellow station arrives. This is ten minutes after the first yellow-bound train. So, in each direction the spacing is every ten minutes, as we said, but for nine minutes out of ten, the next train will be headed toward the red station. With this spacing, Mr. Tsuru will be using the red station on the way home 90% of the time.

math [2]
Math Puzzle [3]