There are three houses, equally spaced on a street, labeled in order A, B, and C, in which five children live. There is at least one child in each house. The children decide on a weekend to arrange a playdate, and the meeting place is the house that minimizes the total distance traveled. What is the probability that the children meet at house B?

Solution: One can track all the possible configurations of children distributed in the three houses, along with the total distance (number of segments) travelled by the children in attending the playdate at the various houses:

We observe that the minimum total distance occurs in house B in 4 out of the 6 cases (highlighted blue), giving a probability of 2/3.

This problem is based on a 1950s Putnam problem, which involved a much more general result. For a shortcut, we note that the house that minimizes the total distance is the one in which where Child #3 resides.