Search: id:A005670
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%I A005670 M3267
%S A005670 1,4,6,7,8,9,9,10,10,11,11,11,11,12,12,12,12,13,13,13,13,13,13,14,
%T A005670 14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,15,16,16,16,16,16,
%U A005670 16,16,16,16,16,16,16,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17
%N A005670 Mrs. Perkins's quilt: smallest coprime dissection of n X n square.
%C A005670 The problem is to dissect an n X n square into smaller integer squares, 
               the gcd of whose sides is 1, using the smallest number of squares. 
               The gcd condition exclude dissecting a 6 X 6 into four 3 X 3 squares.
%C A005670 The name "Mrs Perkins's Quilt" comes from a problem in one of Dudeney's 
               books, wherein he gives the answer for n = 13. I gave the answers 
               for low n and an upper bound of order n^(1/3) for general n, which 
               Trustrum improved to order log(n). There's an obvious logarithmic 
               lower bound. - J. H. Conway, Oct 11, 2003
%D A005670 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005670 J. H. Conway, Mrs. Perkins's quilt, Proc. Camb. Phil. Soc., 60 (1964), 
               363-368.
%D A005670 H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, 
               C3.
%D A005670 Trustrum, G. B., Mrs Perkins's quilt, Proc. Cambridge Philos. Soc., 61 
               1965 7-11.
%H A005670 A. J. W. Duijvestijn, <a href="http://www.squaring.net/downloads/TableI">
               Table I</a>
%H A005670 A. J. W. Duijvestijn, <a href="http://www.squaring.net/downloads/TableII">
               Table II</a>
%H A005670 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/perkinsbestquilts.txt">
               Mrs Perkin's Quilt</a>
%H A005670 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MrsPerkinssQuilt.html">Link to a section of The World of Mathematics.</
               a>
%H A005670 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Mrs.PerkinssQuilt.html">Mrs. Perkins's Quilt</a>
%e A005670 Illustrating a(7) = 9: a dissection of a 7 X 7 square into 9 pieces, 
               courtesy of Ed. Pegg, Jr.:
%e A005670 .___.___.___.___.___.___.___
%e A005670 |...........|.......|.......|
%e A005670 |...........|.......|.......|
%e A005670 |...........|.......|.......|
%e A005670 |...........|___.___|___.___|
%e A005670 |...........|...|...|.......|
%e A005670 |___.___.___|___|___|.......|
%e A005670 |...............|...|.......|
%e A005670 |...............|___|___.___|
%e A005670 |...............|...........|
%e A005670 |...............|...........|
%e A005670 |...............|...........|
%e A005670 |...............|...........|
%e A005670 |...............|...........|
%e A005670 |___.___.___.___|___.___.___|
%e A005670 The Duijvestijn code for this is {{3,2,2},{1,1,2},{4,1},{3}}
%e A005670 Solutions for n=1..10:
%e A005670 1 {{1}}
%e A005670 2 {{1,1},{1,1}}
%e A005670 3 {{2,1},{1},{1,1,1}}
%e A005670 4 {{2,2},{2,1,1},{1,1}}
%e A005670 5 {{3,2},{1,1},{2,1,2},{1}}
%e A005670 6 {{3,3},{3,2,1},{1},{1,1,1}}
%e A005670 7 {{4,3},{1,2},{3,1,1},{2,2}}
%e A005670 8 {{4,4},{4,2,2},{2,1,1},{1,1}}
%e A005670 9 {{5,4},{1,1,2},{4,2,1},{3},{2}}
%e A005670 10 {{5,5},{5,3,2},{1,1},{2,1,2},{1}}
%Y A005670 Cf. A005842.
%Y A005670 Sequence in context: A011275 A006185 A021876 this_sequence A123860 A122817 
               A074764
%Y A005670 Adjacent sequences: A005667 A005668 A005669 this_sequence A005671 A005672 
               A005673
%K A005670 nonn,nice
%O A005670 1,2
%A A005670 N. J. A. Sloane (njas(AT)research.att.com).
%E A005670 Extended using values from Ed Pegg's web site.
%E A005670 It is not clear how many of these terms have been proved to be correct 
               and how many are just conjectures. To be safe, regard all the entries 
               in this sequences as conjectures, unless stated otherwise.

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