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A102508
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Suppose there are chairs around a round table, forming a regular polygon, and there are n persons seated on some of these chairs. Let a(n) be the maximum number of chairs such that, if a waiter puts two glasses (randomly) on the table in front of two (different) chairs, it is always possible to turn the table so that the two glasses end up in front of two seated persons.
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1
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OFFSET
| 2,1
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COMMENTS
| a(n) <= n(n-1)+1. Moreover, a(n)=n(n-1)+1 iff A058241(n)>0, i.e., when a perfect difference set modulo n(n-1)+1 exists. In particular, a(12) = 133, a(14)=183, a(17)=273, etc.
This problem is a circular analogue of an optimal ruler problem; see A004137. - David Wasserman (dwasserm(AT)earthlink.net), Apr 15 2008
Solutions do not always exist for table sizes less than a(n). For example, for n = 5 there is no solution for a table of size 20. - David Wasserman (dwasserm(AT)earthlink.net), Apr 15 2008
95 <= a(11) < 109. - David Wasserman (dwasserm(AT)earthlink.net), Apr 15 2008
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EXAMPLE
| a(5)=21 because if we have 21 chairs, 5 persons can sit down on chairs 1, 4, 5, 10 and 12. 1=5-4 (mod 21). 2=12-10 (mod 21). 3=4-1 (mod 21). 4=5-1 (mod 21). 5=10-5 (mod 21). 6=10-4 (mod 21). 7=12-5 (mod 21). 8=12-4 (mod 21). 9=10-1 (mod 21). 10=1-12 (mod 21). It is impossible to do the same with 22 or more chairs.
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CROSSREFS
| Cf. A004137, A058241.
Sequence in context: A025721 A195020 A169627 * A115298 A161206 A025728
Adjacent sequences: A102505 A102506 A102507 * A102509 A102510 A102511
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KEYWORD
| nonn,more,hard
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AUTHOR
| Ard Van Moer (ard.van.moer(AT)vub.ac.be), Mar 15 2005
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EXTENSIONS
| 3 more terms from David Wasserman (dwasserm(AT)earthlink.net), Apr 15 2008
Edited by Max Alekseyev (maxale(AT)gmail.com), Apr 29 2010
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