Elevated problem
A building has seven elevators that each can stop in not more than six floors. You can reach any floor from any other floor without changing elevator. How many floors can the building have?
Problem source: The Math Forum.
A building has seven elevators that each can stop in not more than six floors. You can reach any floor from any other floor without changing elevator. How many floors can the building have?
Problem source: The Math Forum.
February 5th, 2010 at 12:24 am
Say an elevator connects two floors, or connects a pair of floors, if it stops at both those floors.
Let the number of floors be n.
For each floor F, each elevator that stops at F stops at only 5 other floors, and thus connects 5 pairs that include F. If n is more than 11, then for each floor F there are more than 10 pairs that include F, and thus more than 2 elevators must stop at F.
Each elevator stops at 6 floors, so the 7 elevators colletively stop at 7 * 6 = 42 floors. For each floor F, more than 2 elevators must stop at F, thus there cannot be more than 42/3 = 14 floors.
Example:
Elevator 1 stops at floors 1 2 3
Elevator 2 stops at floors 1 4 5
Elevator 3 stops at floors 1 6 7
Elevator 4 stops at floors 2 4 6
Elevator 5 stops at floors 2 5 7
Elevator 6 stops at floors 3 4 7
Elevator 7 stops at floors 3 5 6
and in addition any elevator that stops at floor n also stops at floor n+7.
The 7 sets of 3 elements from {1, 2, 3, 4, 5, 6, 7} are the 7 lines of the Fano plane.