## 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?