How the Professor Solved His Puzzle While Eating Pizza

An absent-minded professor who lived in midtown Manhattan was affiliated with both Columbia University (uptown) and NYU (downtown). He wanted to have office hours for his students at both universities. Since he lived within walking distance of a subway station that had hourly trains going in each direction, his routine was to go to the station whenever he happened to be ready (at random times), and take the first train that arrived, either the uptown train or the downtown train. In this manner he thought he would, on average, spend the same number of days at both his universities.

After a few weeks, his NYU students began complaining that he was spending 3 times as many days with the Columbia students as with them at NYU. This surprised the absent-minded professor, but after a few more weeks, during which he kept careful records, he discovered that his NYU students had a legitimate complaint. The professor was puzzled.

One day, he went to have lunch at a pizza parlor, where he ordered a small pie that was served to him sliced into quarters. He began eating his first slice while puzzling over his dilemma about his lop-sided office hours between Columbia and NYU. He finished his first slice, and as he reached for his next one, he saw the remaining 3 quarters of his pizza on the serving platter. The solution to his puzzle came to him in a flash! But it was not clear how he might accommodate his NYU students without giving up the luxury of walking to the subway station at random times.

[The solution to the professor's puzzle will be provided after I get some pizza.]

Solution: Although the interval between successive uptown trains, as well as between successive downtown trains, is one hour, the interval between an uptown train and the next downtown train is a quarter hour. Hence, if the professor arrives at random times, the probability that he will board an uptown train is ¾ and the corresponding probability for a downtown train is only ¼.