The building manager of an Amazon warehouse says that that building has
seven elevators and that each elevator can stop and open at only six floors (does
not have the buttons to stop at the remaining floors). She also says that in that
building you can go from any floor to any other floor by taking only one elevator
(provided you take the correct one) so that the people who work in the
warehouse do not waste time taking multiple elevators.
What is the largest number of floors that such a building can have?


short resuemn

The problem describes a building with seven elevators, each serving only six floors, and requires that you can travel between any two floors using a single elevator. This is a combinatorial problem related to graph theory, where each elevator connects a subset of floors, and every pair of floors must be connected by at least one elevator.

To solve this, the elevators and floors can be thought of as a graph with vertices (floors) and edges (elevators that connect floors). This leads to a situation where the system should cover all possible pairs of floors with the given constraints.

The largest number of floors is **13**, which is achieved by carefully choosing which elevators serve which floors so that each pair of floors can be connected by at least one elevator.

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Do you want further details or have any questions?

Here are some related questions:
1. What is the mathematical principle used to determine the largest number of floors?
2. Can you explain how the graph theory concept applies to this problem?
3. Are there other real-life examples where a similar problem arises?
4. How would the solution change if each elevator could stop at more or fewer floors?
5. How do combinatorics help in solving problems involving coverage with restrictions?

**Tip:** Understanding graph theory helps in solving connectivity problems like this!

The building manager of an Amazon warehouse says that that building has seven elevators and that each elevator can stop and open at only six floors (does not have the buttons to stop at the remaining floors). She also says that in that building you can go from any floor to any other floor by taking only one elevator (provided you take the correct one) so that the people who work in the warehouse do not waste time taking multiple elevators. What is the largest number of floors that such a building can have?

Discover how to solve the seven-elevator problem where each elevator serves six floors, and any two floors must be reachable by a single elevator. This problem explores key concepts in graph theory and combinatorics, applying the graph connectivity theorem to determine that the largest number of floors such a building can have is 13. Suitable for undergraduate students and math enthusiasts.

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