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Problem of the Month (November 2009)

This month we examine a famous problem involving elevators. Suppose there are e elevators in a building, and each one stops at s of the floors. We call a building convenient if you can get from any floor to any other floor without changing elevators. What is the maximum number of floors f(e,s) of a convenient building? For example, f(4,3)≤5 since the 4 elevators could stop at floors {1,2,3}, {1,2,4}, {1,2,5}, and {3,4,5}. To see that f(4,3)<6, notice the elevators only make 4×3=12 stops, so some floor will only be visited by at most 2 elevators, and that floor will not be accessible from every floor. What are the values of f(e,s)? What can be proven about the function f(e,s)?

We call a building perfect if each pair of floors is linked by exactly one elevator. The only perfect buildings known are the trivial e=1 and s=k, and e=k(k-1)/2 and s=2, and the non-trivial pairs e=7 and s=3, e=12 and s=3, and e=13 and s=4. Can you find the designs of these perfect buildings? Can you find any others? Are there many other perfect buildings if we relax the rule that every elevator has to visit the same number of floors, but insist that every elevator visits at least 3 floors?

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