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:

    Man, in the ideal, is so noble and so sparkling, such a grand and glowing creature, that over any ignominious blemish in him all his fellows should run to throw their costliest robes.
    Herman Melville (1819–1891)

    In soliciting donations from his flock, a preacher may promise eternal life in a celestial city whose streets are paved with gold, and that’s none of the law’s business. But if he promises an annual free stay in a luxury hotel on Earth, he’d better have the rooms available.
    Unknown. Charlotte Observer (October 6, 1989)

    A woman is only a woman, but a good cigar is a smoke.
    Jo Swerling (b. 1897)

    Yet there is a mystery here and it is not one that I understand: without the sting of otherness, of—even—the vicious, without the terrible energies of the underside of health, sanity, sense, then nothing works or can work. I tell you that goodness-what we in our ordinary daylight selves call goodness: the ordinary, the decent—these are nothing without the hidden powers that pour forth continually from their shadow sides. Their hidden aspects contained and tempered.
    Doris Lessing (b. 1919)