December 2005


Brainteaser -  Hotel Infinity

Suppose that, somewhere in New Jersey, there is a hotel with an infinite number of rooms.  You arrive late one night and ask the front desk clerk if they have a vacancy.  He replies that every room is occupied, however he can arrange for you to get one.  But how, you wonder, if there is no vacancy? 


The answer is simple:  the clerk will simply ask the people in room 1 to move to room 2, those in room 2 to move to room 3, those in 3 to move 4, and so on.  Since there is an infinite number of rooms, everyone will have a room to move into, and room 1 will be available for you.


Hotel Infinity is an amazing place, you think to yourself as you sign in.  But just as the clerk is about to give you your key, an infinite number of people arrive for an APA convention.  The clerk cleverly figured out how to get you a room, but can he accommodate an additional infinity of guests?  Amazingly, he can. 


He just asks everyone to move again, but this time to the room number that is twice the number of their current room.  In other words, you would move to room 2, the people in 2 would move to 4, those in 3 to 6, those in 4 to 8 and so on.  This will leave all odd numbered rooms - an infinite number of them - vacant.


The Moral:  We tend to think of marketplaces as finite.  This finite view has established companies continuously surprised at the ability of new entrants to build substantial market share.  When we think of marketplaces as infinite, we recognize that there is always room for another player.


Thanks to the folks at for cataloging this brainteaser.


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