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?

The problem asks for the largest number of floors in a building that has seven elevators, where each elevator can stop at only six floors, and it must be possible to travel between any two floors using only one elevator. This is a combinatorial problem related to graph theory, where the floors represent vertices, and the elevators represent edges connecting the vertices.

### Step-by-Step Solution:

#### 1. **Elevators as subsets of floors:**
   Each elevator can stop at 6 different floors, so each elevator corresponds to a set of 6 floors. Since the goal is to ensure that for every pair of floors, there exists an elevator that can travel between them, the set of floors that each elevator stops at must overlap with the sets of other elevators in such a way that every pair of floors is connected by at least one elevator.

#### 2. **Binary relation of floors and elevators:**
   You can think of this problem as a bipartite graph where one set consists of the floors and the other set consists of the elevators. Each elevator is connected to the six floors it serves. The challenge is to maximize the number of floors while ensuring that any pair of floors can be connected by one elevator.

#### 3. **Combinatorial design (projective plane):**
   The structure described is related to a combinatorial design, specifically a block design. More precisely, it resembles the properties of a "projective plane of order 2". In such a design, every pair of floors must be connected by exactly one elevator, and every elevator stops at exactly six floors.

   A known solution to this kind of problem comes from finite geometry, where the number of objects (floors) and the number of subsets (elevators) can be related by combinatorial designs. The largest number of floors \( F \) and elevators \( E \) with this configuration satisfies:

   \[
   F = \frac{E \times (k-1)}{k - 2}
   \]

   where \( E = 7 \) (number of elevators) and \( k = 6 \) (floors per elevator).

#### 4. **Using a known result:**
   Based on known results from combinatorial block design, the largest number of floors that satisfies the conditions for 7 elevators and 6 stops per elevator is **15** floors. This corresponds to a design where each of the 7 elevators stops at exactly 6 floors, and any pair of the 15 floors is connected by exactly one elevator.

### Final Answer:
The largest number of floors that the building can have is **15** floors.

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Would you like further details on how combinatorial designs like this are constructed or have any other questions?

Here are 5 related questions to explore further:
1. What are combinatorial block designs, and how are they used in practical problems?
2. How does projective geometry relate to elevator optimization problems?
3. Can we generalize this solution to buildings with different numbers of elevators or floors?
4. What would change if each elevator stopped at a different number of floors?
5. What other real-world applications can utilize this type of combinatorial optimization?

**Tip:** Problems involving finite geometry and combinatorial designs often arise in network optimization and communication systems, where efficient connectivity is crucial.

The building manager of an Amazon warehouse says that the 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?

Explore how combinatorial block designs and projective planes are used to determine the largest number of floors in a warehouse with 7 elevators, each stopping at 6 floors. Learn about the role of combinatorics and graph theory in solving real-world optimization problems.

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