## Math Brain Teaser: Unfinished Thesis

You are spending the summer polishing your thesis in the university library. Every day you take the escalator into the subway, turn right and catch a train going up North to the university. One day you realize that the trains on your left going in the opposite direction can bring you to the beach. It is summer and nothing is wrong with some leisure. You carefully calculate that even if you spend half of the remaining summer vacation in the library it should be enough to finish the thesis. You decide to spice up your summer by allowing some chance to guide your life: every day you catch the first train that comes to the platform. It may be train on the left going to the beach or train on the right heading to the university.

You think that because you wake up and come to the platform randomly in-between 9 and 10am, and the trains go on a regular schedule, with the same frequency in both directions, you should end up spending about the same amount of time in the library and at the beach. On the first week of this experiment you are surprised that chance brought you to the library only once a week. “I know, it will correct itself on a long run, like heads-and-tails game” – you say and continue with the same strategy. But even after two months you find yourself at the library only 1/5th of the time. How could this happen if you do not cheat?

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Puzzle Answer: It is all defined by the train schedule. The trains DO go on a regular schedule, with the same frequency in both directions. Let’s assume trains leave every 10 mins, with train to the beach departing from your station at 10 minutes past the hour, 20 mins, 30 mins etc while train to the library leaves exactly 2 mins after the beach train, at 12, 22, 32,.. mins past the hour. Then, only if you come in-between 10 and 12, 20 and 22, 30 and 32 mins past the hour you you will wait and pick the library train. If you come in-between 12 and 20, 22 and 30, 32 and 40 mins past the hour you will wait and pick the beach train. 2 minute windows for the library train and 8 minute windows for the beach train. As you arrive to the platform randomly in-between 9 and 10am, your chances of seeing the library train first are 2/10=1/5 and beach train first are 8/10=4/5. So, you end up finding yourself at the library on average only once out of 5-day work week and at the beach 4 times during the 5-day work week. Blame the train schedule or beware that random doesn’t always lead to half-and-half!

PS: Enjoy these 50 brain teasers to test your cognitive ability. Free, and fun for adults of any age!