Hilbert's Paradox of The Grand Hotel - The Grand Hotel Cigar Mystery

The Grand Hotel Cigar Mystery

Another story regarding the Grand Hotel can be used to show that mathematical induction only works from an induction basis.

Suppose that the Grand Hotel does not allow smoking, and no cigars may be taken into the Hotel. Despite this, the guest in room 1 goes to the guest in room 2 to get a cigar. The guest in room 2 goes to room 3 to get two cigars—one for himself and one for the guest in room 1. In general, the guest in room N goes to room (N+1) to get N cigars. They each return, smoke one cigar and give the rest to the guest from room (N-1). Thus despite the fact no cigars have been brought into the hotel, each guest can smoke a cigar inside the property.

The fallacy of this story derives from the fact that there is no inductive point (base-case) from which the induction can derive. Although it is shown that if the guest from room N has N cigars then both he and all guests in lower-numbered rooms can smoke, it is never proved that any of the guests actually have cigars therefore it doesn't follow that any guest can smoke a cigar inside the Hotel. The fact that the story mentions that cigars are not allowed into the hotel is designed to highlight the fallacy. However, since there is an infinite number of rooms in the hotel and each guest (N) must go to guest (N+1) for his cigar, this process of going up one room never ends and no cigars are ever smoked.

Read more about this topic:  Hilbert's Paradox Of The Grand Hotel

Famous quotes containing the words grand, hotel, cigar and/or mystery:

    The evolution of humans can not only be seen as the grand total of their wars, it is also defined by the evolution of the human mind and the development of the human consciousness.
    Friedrich Dürrenmatt (1921–1990)

    The hotel was once where things coalesced, where you could meet both townspeople and travelers. Not so in a motel. No matter how you build it, the motel remains the haunt of the quick and dirty, where the only locals are Chamber of Commerce boys every fourth Thursday. Who ever heard the returning traveler exclaim over one of the great motels of the world he stayed in? Motels can be big, but never grand.
    William Least Heat Moon [William Trogdon] (b. 1939)

    Oh well,
    it doesn’t belong to me,
    if a cigar can be a cigar
    then a dream can be a dream.
    Right?
    Anne Sexton (1928–1974)

    A proper secrecy is the only mystery of able men; mystery is the only secrecy of weak and cunning ones.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)