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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?

(Puz­zle answer is below. Please try to solve it before check­ing out the answer!)

Maria Lando, MathmomMaria Lan­do (aka The­Math­Mom) illu­mi­nates and demys­ti­fies math­e­mat­ics for adults, describ­ing its fun use in every­day life. The­Math­Mom has been fea­tured in the Boston Globe, The Jew­ish Advo­cate, and Red­book Mag­a­zine. Her week­ly newslet­ter con­tains sto­ries and puz­zles for the whole fam­ily, and tips on per­ceiv­ing math and pre­sent­ing it to kids as a toy, a tool, and a friend. Learn more at

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!

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