There is an east-west street of length L units. And we park cars of unit length along the north side until we can't place any more cars. Each car is placed randomly (uniformly).

What is the expected number of cars that can be parked (as a function of L)?
__________________________

I'll start you off...

For 0 <= L < 1, F(L) = 0
For 1 <= L < 2, F(L) = 1

Okay... now the easy ones are out of the way, can you describe the function for L>=2?

(In reply to simulation by Charlie)

"Toward the end of this series, 10 extra units seems to allow 7.5 additional cars, so possibly the average number of cars approaches 75% of the length of the curb as that length increases without limit."

Close, but no cigar.  Oh that the world could always give us nice answers like 0.75.  But then a Hungarian mathematician would not have gotten a constant named after him. The exact limit is Renyi's Parking Constant which according to Simon Plouffe (see http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap71.html ) is equal to 0.74759792025341143517873094363652421026172... .  See the previous comment for a reference that leads to a number of others as well.

Edited on July 7, 2004, 1:47 am

Posted by Richard on 2004-07-05 19:49:36