Guests at a Round Table
Fifteen chairs are evenly placed around a circular table. On the table are the name cards of fifteen guests. After the guests sit down, it turns out that none of them is sitting in front of his own card. Prove that the table can be rotated so that at least 2 guests are simultaneously correctly sitted.
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Copyright © 1996-2018 Alexander Bogomolny
Fifteen chairs are evenly placed around a circular table. On the table are the name cards of fifteen guests. After the guests sit down, it turns out that none of them is sitting in front of his own card. Prove that the table can be rotated so that at least 2 guests are simultaneously correctly sitted.
The table has 15 possible positions. One of this positions, viz., the initial one, has no man matching a card, leaving 14 positions for possible matchings. There are 15 people and 14 table positions. By the pigeonhole principle, there bound to be a position for which at least to men match their names.
Clearly, the result holds for any number of guests.
References
- A Decade of the Berkeley Mathematical Circle, The American Experience, Volume I, Z. Stankova, Tom Rike (eds), AMS/MSRI, 2008, pp. 165-168
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