Math puzzle: Can you meet me at the mall?
You and a friend have arranged to meet at a popular downtown mall between 3 p.m. and 4 p.m. one afternoon. However, you neglected to specify a time within that one-hour window. Therefore, each of you will be arriving at randomly selected times between 3 p.m. and 4 p.m. Once each of you arrives at the mall, you will be there for exactly 15 minutes. When the 15 minutes are up, you leave.
- During the hour, there may or may not be an overlap between your and your friend’s visits. At some point, how many of you are present will reach a maximum number for the hour. This maximum could be one (sad!) or two. On average, what do you expect this maximum to be? The answer is between one and two.
- Hint: If you’re not sure where to start, think of the two arrival times on a coordinate plane. Your arrival time is the x-coordinate and your friend’s is the y. Which region in the coordinate plane are you considering in this puzzle? Which region results in the two of you meeting up?
- Instead of you and a friend, now suppose there are three total friends, yourself included. As before, you and the friends arrive at random times during the hour and each stay for 15 minutes. Again, at some point during the hour, there will be a maximum number of friends at the mall. This maximum could be one, two or three. On average, what would you expect this maximum number of friends to be?
- What about four total friends? On average, what would you expect the maximum number of friends meeting up to be?
- Hint: If you can’t find the exact answer, try finding an estimate. A computer might help.
- Suppose there are N friends. As N grows increasingly large, what would you expect the maximum number of friends meeting up to be, in terms of N?
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