Search: id:A066385
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%I A066385
%S A066385 6,9,10,11,14,15,16,18,20,21,23,24,25,27,29,30,32,33,35,36,38,39,41,42,
%T A066385 44,45,47,48,50,51,53,54,56,57,59,60
%N A066385 Smallest maximum of sum of 3 consecutive terms in any arrangement of 
               [1..n] in a circle.
%C A066385 In a problem in the "Bundeswettbewerb 2002" competition there are 12 
               sticks of lengths 1,..,12 put in a ring in random order. It has to 
               be proved that there are at least 3 consecutive sticks with total 
               length not less than 20. A closer look shows that the total length 
               is at least a(12)=21. The problem of the contest is a consequence 
               of the following observation: every term a(n) is at least ceil(3*(n+1)/
               2), since n*a(n) >= sum{i=1..n}(p(i-1)+p(i)+p(i+1)) = 3*sum{i=1..n}(i) 
               =3*n*(n+1)/2. So in the case n=12 we have (total length) >= a(12)=21 
               >= 20.
%D A066385 Thread "Zahlenkreis" in de.sci.mathematik, December 2001
%D A066385 T. J. Rolfe, P. W. Purdom, "An Alternative Problem for Backtracking and 
               Bounding", Bulletin of the ACM SIG on Computer Science Education, 
               Vol. 36, No. 4 (December 2004), pp. 83-84 [From Milan Stefanovic 
               (stefke381(AT)gmail.com), Nov 19 2008]
%H A066385 <a href="http://www.bundeswettbewerb-mathematik.de/aufgaben/aufgaben.htm">
               Bundeswettbewerb Mathematik 2002 (1.4)</a>
%F A066385 Let p be a permutation of 1..n and let g(p) be the maximum of the consecutive 
               triple sums p(i-1)+p(i)+p(i+1), where p(0)=p(n) and p(n+1)=p(1). 
               a(n) is the minimum of all the g(p) taken over all permutations p.
%e A066385 a(6)=11 because cycle 1-4-5-2-3-6- has sums 11,10,11,10,11,10 with max=11.
%e A066385 This example by Helmut Richter shows that a(14) = 24 is very likely: 
               p = (1-8-11-4-9-10-2-12-5-6-13-3-7-14-) with g(p) = 11+4+9 = 24 as 
               maximal three-sum.
%Y A066385 Sequence in context: A140052 A070598 A124257 this_sequence A103092 A104523 
               A091886
%Y A066385 Adjacent sequences: A066382 A066383 A066384 this_sequence A066386 A066387 
               A066388
%K A066385 nice,nonn
%O A066385 3,1
%A A066385 Rainer Rosenthal (r.rosenthal(AT)web.de), Dec 23 2001
%E A066385 Terms a(1) - a(20) are from the cited reference. The rest, a(21) - a(38) 
               are obtained using the program from the same reference. Milan Stefanovic 
               (stefke381(AT)gmail.com), Nov 19 2008
%E A066385 Broken link corrected by Rainer Rosenthal (r.rosenthal(AT)web.de), Jun 
               18 2009

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