Math puzzle: Can you meet me at the mall?

You and a chum hold organized to meet at a most in type downtown mall between 3 p.m. and 4 p.m. one afternoon. However, you neglected to specify a time inside of that one-hour window. Therefore, every of you are going to be arriving at randomly selected instances between 3 p.m. and 4 p.m. As soon as every of you arrives at the mall, you are going to be there for exactly Quarter-hour. When the Quarter-hour are up, you whisk away.

1. All the way thru the hour, there may or are not an overlap between your and your buddy’s visits. At some level, what number of of you are fresh will attain a most amount for the hour. This most might be one (unhappy!) or two. On common, what attain you ask this most to be? The reply is between one and two.
   • Hint: As soon as you’re not distinct the attach to launch, mediate of the 2 arrival instances on a coordinate plane. Your arrival time is the x-coordinate and your buddy’s is the y. Which field in the coordinate plane are you bearing in tips on this puzzle? Which field ends in the 2 of you assembly up?
2. Somewhat than you and a chum, now notify there are three total mates, yourself integrated. As earlier than, you and the mates come at random instances in the route of the hour and each give up for Quarter-hour.
   • Hint: Again, at some level in the route of the hour, there'll likely be a most amount of mates at the mall. This most might be one, two or three. On common, what would you ask this most amount of mates to be?
3. What about four total mates? On common, what would you ask basically the most amount of mates assembly up to be?
   • Hint: As soon as you might’t accumulate the particular reply, strive discovering an estimate. A computer can lend a hand.
4. Explain there are N mates. As N grows more and more great, what would you ask basically the most amount of mates assembly up to be, when it comes to N?

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