## Ken's POTW

Five Floor Elevators
The owner of the Party Hotel chain, Happy Harold, enjoyed a very good business, renting his entire hotel for group parties (he did especially well during the Spring vacations of colleges.) Harold also owned an elevator business and (of course) supplied his hotel with his own elevators. Harold was a bit frugal, however, and only put six elevators in his hotel. To make matters worse, his elevator designers hadn't passed the demonstration phase of their designs before Harold put them to use and each elevator could stop at only five floors (the floors weren't necessarily continuous - the elevators were limited because only that many doors shipped with the demonstration elevators).
1. Harold's guests found it an intriguing aspect of the hotel that no more than two elevator rides were needed to reach any floor of the hotel. What was the maximum number of floors in the hotel? Can you show a possible configuration for the elevators?

2. Harold's guests said it sometimes took too long to wait for an elevator on the floors that only had one elevator stop. Harold built his next hotel such that every floor had two elevators stop at it, and no two elevators stopped at the same two floors. Again, no more than two elevator rides were needed to reach any other floor. What is the maximum number of floors in this new hotel? Can you show a possible configuration for the elevators?

3. Although the guests found the multiple elevator rides unique, they eventually tired of needing to switch. Harold built his next hotel such that only one elevator ride was necessary to reach any other floor. What is the maximum number of floors in this new hotel? Can you show a possible configuration for the elevators?

Extra Credit: Harold realized that his profits kept getting smaller as his hotels did, but he never invested in more elevator doors. How many elevators would he need to build to the height of his first hotel, but meet the requirements of the last?

Source: Original.

Solution
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