Math Problem of the Fortnight: Dedekind’s Diner
The town of Cantoria has many residents—infinitely many, in fact—who are very patient. This is fortunate, because the only restaurant in town, Dedekind’s Diner (across the street from Hilbert’s Hotel), has only one table, so patrons usually have to wait in a line. When people arrive at the diner, they take a number and wait until their number is called.
One day, at 10:00 in the morning, ten people arrive at Dedekind’s Diner to eat. They take the numbers 1 through 10. The table is available, so person 1 sits down and eats. He is a rather slow eater, so it takes him one hour to finish his meal.
At 11:00, ten more people arrive at the diner, taking the numbers 11 through 20. Person 1 gets up from the table, pays his bill, and leaves, so person 2 sits down and eats. She eats faster than person 1, finishing her meal in half an hour.
At 11:30, ten more people arrive, taking the numbers 21 through 30. Person 2 finishes her meal, pays her bill, and leaves, and person 3 sits down and eats. He is even faster than person 2; he takes only 15 minutes to eat his meal.
This pattern continues. Each person takes half as long to eat his or her meal as the previous person did; when he or she gets done, ten more people arrive at the diner and take the next ten numbers, and the next person in line sits down to eat.
At the stroke of noon, how many people are waiting to eat?
Winner
The winning solution was by Seth Hoffert.
