Brain teaser: Will you finish your thesis on time?

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Imag­ine spend­ing the sum­mer pol­ish­ing your PhD the­sis in the library. Every day you take the sub­way, a train going up North to the university.

One day you real­ize that the trains going South can bring you to the beach. Noth­ing is wrong with some leisure. You cal­cu­late that if you spend half of the remain­ing sum­mer vaca­tion in the library that should be enough to fin­ish the the­sis. To spice up the sum­mer,every day you catch the first train that comes to the plat­form. It may be South train going to the beach or the North one head­ing to the university.

Since you 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 guessti­mate you will 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. “This will cor­rect itself dur­ing the sum­mer, like a heads-and-tails game,” you think. But even after two months you find your­self at the library only 1/5th of the time– and the the­sis at grave risk.

How could this happen?Maria Lando, Mathmom

(The answer is below. For your brain’s sake, try to solve it before check­ing out the answer!)

Maria Lan­do (aka The­Math­Mom) illu­mi­nates and demys­ti­fies math­e­mat­ics for adults, describ­ing its fun use in every­day life. The­Math­Mom has been fea­tured in mul­ti­ple media out­lets includ­ing The Boston Globe, The Jew­ish Advo­cate, and Red­book Mag­a­zine. Learn more at TheMathMom.com.

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SOLUTION

Sad­ly, you ignored one cru­cial variable.

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,.. min­utes 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.

Ran­dom does­n’t always lead to half-and-half 🙂

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