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Brain teaser: Will you finish your thesis on time?


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



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 🙂

More brain teaser games for teens and adults to challenge your cognitive skills:

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As seen in The New York Times, The Wall Street Journal, BBC News, CNN, Reuters,  SharpBrains is an independent market research firm tracking how brain science can improve our health and our lives.

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