20161130, 22:06  #12  
"Robert Gerbicz"
Oct 2005
Hungary
1429_{10} Posts 
Quote:
Optimal solutions for n=29,30,31,32: Code:
a(29)=9 aaaaaaaaaaaaaaaaaaaaaaaaabbbb aaaaaaaaaaaaaaaaaaaaaaaaabbbb aaaaaaaaaaaaaaaaaaaaaaaaabbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccddddddddeeeeeeeeefffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggiiiiiiiiiiifffffbbbb cccggggggjjjjjjjjjjjjkkkkkkkk cccggggggjjjjjjjjjjjjkkkkkkkk cccggggggjjjjjjjjjjjjkkkkkkkk cccggggggjjjjjjjjjjjjkkkkkkkk cccggggggjjjjjjjjjjjjkkkkkkkk cccggggggjjjjjjjjjjjjkkkkkkkk ccchhhhhhhhhhhhhhhhhhkkkkkkkk ccchhhhhhhhhhhhhhhhhhkkkkkkkk ccchhhhhhhhhhhhhhhhhhkkkkkkkk ccchhhhhhhhhhhhhhhhhhkkkkkkkk a(30)=11 aaaaaaaaaaaaaaaaaaaaaaaaaaaabb aaaaaaaaaaaaaaaaaaaaaaaaaaaabb cccccccccccddddddddddddddeeebb cccccccccccddddddddddddddeeebb cccccccccccddddddddddddddeeebb cccccccccccddddddddddddddeeebb cccccccccccffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkkmmmmmmmmffffgggggghhhheeebb kkknnnnnnnnffffgggggghhhheeebb kkknnnnnnnnffffgggggghhhheeebb kkknnnnnnnnffffiiiiiihhhheeebb kkknnnnnnnnffffiiiiiihhhheeebb kkknnnnnnnnffffiiiiiihhhheeebb kkknnnnnnnnffffiiiiiihhhheeebb kkknnnnnnnnffffiiiiiihhhheeebb kkknnnnnnnnffffiiiiiijjjjjjjbb kkkooooooooooooiiiiiijjjjjjjbb kkkooooooooooooiiiiiijjjjjjjbb kkkooooooooooooiiiiiijjjjjjjbb kkkooooooooooooiiiiiijjjjjjjbb kkkooooooooooooiiiiiijjjjjjjbb llllllllllllllllllllljjjjjjjbb llllllllllllllllllllljjjjjjjbb llllllllllllllllllllljjjjjjjbb a(31)=11 aaaaaaaaaaaaaaaaaaaaaaaaabbbbbb aaaaaaaaaaaaaaaaaaaaaaaaabbbbbb cceeeeeeeeeeeeeeeeeeeeeeebbbbbb cceeeeeeeeeeeeeeeeeeeeeeebbbbbb ccffgggggggggghhhhhhhhhhhbbbbbb ccffgggggggggghhhhhhhhhhhbbbbbb ccffgggggggggghhhhhhhhhhhbbbbbb ccffgggggggggghhhhhhhhhhhbbbbbb ccffgggggggggghhhhhhhhhhhbbbbbb ccffiiiiiiiiiiiiiiiiiiiiiiiiiii ccffiiiiiiiiiiiiiiiiiiiiiiiiiii ccffjjjkkkkkkkkkkkkkkkkllllllll ccffjjjkkkkkkkkkkkkkkkkllllllll ccffjjjkkkkkkkkkkkkkkkkllllllll ccffjjjmmmmnnnnnnnnnnnnllllllll ccffjjjmmmmnnnnnnnnnnnnllllllll ccffjjjmmmmnnnnnnnnnnnnllllllll ccffjjjmmmmnnnnnnnnnnnnllllllll ccffjjjmmmmoooooooooooooooppppp ccffjjjmmmmoooooooooooooooppppp ccffjjjmmmmoooooooooooooooppppp ccffjjjmmmmqqqqqqqrrrrrrrrppppp ccffjjjmmmmqqqqqqqrrrrrrrrppppp ccffjjjmmmmqqqqqqqrrrrrrrrppppp ccffjjjmmmmqqqqqqqrrrrrrrrppppp ccffjjjmmmmqqqqqqqrrrrrrrrppppp ccffjjjmmmmqqqqqqqrrrrrrrrppppp ccffjjjmmmmqqqqqqqsssssssssssss ddddddddddddddddddsssssssssssss ddddddddddddddddddsssssssssssss ddddddddddddddddddsssssssssssss a(32)=10 aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbb aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbb aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbb aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbb aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbb aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbb aaaaaaaaaaaaaacccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffggggggggggcccccccccdddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhiiiiiiiiiiiidddddeeee ffffhhhhhhhjjjjjjjjjjjjjjjjjeeee ffffhhhhhhhjjjjjjjjjjjjjjjjjeeee ffffhhhhhhhjjjjjjjjjjjjjjjjjeeee ffffhhhhhhhjjjjjjjjjjjjjjjjjeeee ffffhhhhhhhjjjjjjjjjjjjjjjjjeeee ffffhhhhhhhjjjjjjjjjjjjjjjjjeeee 

20161201, 02:29  #13 
Romulan Interpreter
Jun 2011
Thailand
3×3,049 Posts 
Wow, these are very nice solutions for a very nice puzzle!
Congrats R. Gerbicz! Question: when you say is optimal, does it means that you made exhaustive search? It somehow escapes me how this can be done efficiently... 
20161201, 03:10  #14 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
I know I said most of what I know was useless but partitions are key so is the fact that a number can appear only roughly half the times it shows up on the multiplication table up to n times n. so for example for n= 18 there are roughly 123 values of your square size that are distinct in area ( some can obviously show up more than once) you can then use those and other things potentially to limit the number of partitions to check through. like all 1's is not possible any with primes more than once are also out, so are any with primes greater than the side length. etc.

20161201, 05:12  #15 
Nov 2016
2^{4} Posts 
I've posted an update of my own results to 999 at
http://demonstrations.wolfram.com/MondrianArtProblem/ My method for these high numbers is to create perfect Blanche dissections using some of the methods from Squaring.net. These nonrational solutions are then rounded to integers 10 to 999, and the improvements are noted. All results are loosely based on valence3 graphs, but there are plenty of solutions from order2 graphs. I have added a variant  a rectangle can be repeated if it has a different orientation. For n=3 to 25, my best results are x, x, 2, 2, 4, 3, 3, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 7 Improvements will be greatly appreciated. I haven't yet incorporated the new results for 29 to 31. Last fiddled with by EdPeggJr on 20161201 at 05:12 
20161201, 21:35  #16 
"Robert Gerbicz"
Oct 2005
Hungary
1429_{10} Posts 
Yes, these has been found with an exhaustive search, hence there is no better solution. Still working on some larger n values.

20161201, 23:06  #17 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Code:
my(a=matrix(n,n,i,j,i*j),b=[],c=if(n%2,n,2*n)); for(x=1,#a, if(#x==1, b=a[x,], b=setunion(b,a[x,]) ) ); a=apply(r>select(q>q<19&&q>(r/19),divisors(r)),b); forpart(x=n^2, y=Vec(x); if(y[#y]y[#1]>0 && (y[#y]y[1]==cy[#y]y[1]<c)&& setminus(y,b)==[]&& select(r>2*#select(q>q==r,y)<numdiv(r),y)==y&& setminus([n],fold((x,y)>setbinop((x,y)>x+y,x,y),apply(r>a[setsearch(b,r)],y)))==[], c=y[#y]y[1];print(y) ) , , 17 ) 
20161202, 11:12  #18 
Dec 2016
Germany
1 Posts 
Hi there!
After Mr. Pegg pointed out to me, that all the fun considering this problem takes place over here, I couldn't resist to register immediately. During the last weeks I also worked on an algorithm, that exhaustively searches for an optimal solution. I agree with all solutions, R. Gerbicz found. Even no solution exists with fewer rectangles. I'd like to add the following two solutions (while others are still workinprogress): a(37)=11 Code:
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbfffflll bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbfffflll bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbfffflll dddddddddddddddjjjjjjeeeeeeeeefffflll dddddddddddddddjjjjjjeeeeeeeeefffflll dddddddddddddddjjjjjjeeeeeeeeefffflll dddddddddddddddjjjjjjeeeeeeeeefffflll dddddddddddddddjjjjjjeeeeeeeeefffflll dddddddddddddddjjjjjjeeeeeeeeefffflll nnnngggggggggggjjjjjjeeeeeeeeefffflll nnnngggggggggggjjjjjjeeeeeeeeefffflll nnnngggggggggggjjjjjjeeeeeeeeefffflll nnnngggggggggggjjjjjjeeeeeeeeefffflll nnnngggggggggggjjjjjjmmmmmmmmmfffflll nnnngggggggggggjjjjjjmmmmmmmmmfffflll nnnngggggggggggjjjjjjmmmmmmmmmfffflll nnnngggggggggggjjjjjjmmmmmmmmmfffflll nnnniiiiiiiiiiiiiiiiimmmmmmmmmfffflll nnnniiiiiiiiiiiiiiiiimmmmmmmmmfffflll nnnniiiiiiiiiiiiiiiiimmmmmmmmmfffflll nnnniiiiiiiiiiiiiiiiimmmmmmmmmfffflll nnnniiiiiiiiiiiiiiiiimmmmmmmmmfffflll nnnnkkkkkkkkkkkkcccccccccccccccccclll nnnnkkkkkkkkkkkkcccccccccccccccccclll nnnnkkkkkkkkkkkkcccccccccccccccccclll nnnnkkkkkkkkkkkkcccccccccccccccccclll nnnnkkkkkkkkkkkkcccccccccccccccccclll nnnnkkkkkkkkkkkkaaaaaaaaaaaaapppppppp nnnnkkkkkkkkkkkkaaaaaaaaaaaaapppppppp ooooooooooooooooaaaaaaaaaaaaapppppppp ooooooooooooooooaaaaaaaaaaaaapppppppp ooooooooooooooooaaaaaaaaaaaaapppppppp ooooooooooooooooaaaaaaaaaaaaapppppppp ooooooooooooooooaaaaaaaaaaaaapppppppp hhhhhhhhhhhhhhhhhhhhhhhhhhhhhpppppppp hhhhhhhhhhhhhhhhhhhhhhhhhhhhhpppppppp hhhhhhhhhhhhhhhhhhhhhhhhhhhhhpppppppp Code:
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiiii aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiiii aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiiii aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggccccccccccccccckkkkkkkkkkkkkkiiii gggggeeeeeeeeeeeeeeeeeeeeeedddddddiiii gggggeeeeeeeeeeeeeeeeeeeeeedddddddiiii gggggeeeeeeeeeeeeeeeeeeeeeedddddddiiii gggggeeeeeeeeeeeeeeeeeeeeeedddddddiiii gggggeeeeeeeeeeeeeeeeeeeeeedddddddiiii gggggeeeeeeeeeeeeeeeeeeeeeedddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii gggggffffffffffffhhhhhhhhhhdddddddiiii bbbbbbbbbbbbbbbbbhhhhhhhhhhdddddddiiii bbbbbbbbbbbbbbbbbhhhhhhhhhhdddddddiiii bbbbbbbbbbbbbbbbbjjjjjjjjjjjjjjjjjjjjj bbbbbbbbbbbbbbbbbjjjjjjjjjjjjjjjjjjjjj bbbbbbbbbbbbbbbbbjjjjjjjjjjjjjjjjjjjjj bbbbbbbbbbbbbbbbbjjjjjjjjjjjjjjjjjjjjj bbbbbbbbbbbbbbbbbjjjjjjjjjjjjjjjjjjjjj bbbbbbbbbbbbbbbbbjjjjjjjjjjjjjjjjjjjjj 
20161202, 15:10  #19 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
okay I found one error on my own like the 19's instead of n+1. still a lot to think about since my function fails for n=3 etc still.

20161202, 15:40  #20  
"Robert Gerbicz"
Oct 2005
Hungary
1,429 Posts 
Quote:
I've found with my improved code a(33)a(44) and a larger one: a(47), these are (showing also my solution for n=37,38) Code:
a(33)=12 aaaaaaaaaaaaaaaaaaaaccccccccbbbbb aaaaaaaaaaaaaaaaaaaaccccccccbbbbb aaaaaaaaaaaaaaaaaaaaccccccccbbbbb aaaaaaaaaaaaaaaaaaaaccccccccbbbbb eeeiiihhhhhhhhhhhhhhccccccccbbbbb eeeiiihhhhhhhhhhhhhhccccccccbbbbb eeeiiihhhhhhhhhhhhhhccccccccbbbbb eeeiiihhhhhhhhhhhhhhccccccccbbbbb eeeiiihhhhhhhhhhhhhhccccccccbbbbb eeeiiihhhhhhhhhhhhhhccccccccbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmgggggggggggbbbbb eeeiiilllllmmmmmmffffffffffffffff eeeiiilllllmmmmmmffffffffffffffff eeeiiilllllmmmmmmffffffffffffffff eeeiiilllllmmmmmmffffffffffffffff eeeiiilllllmmmmmmffffffffffffffff eeeiiillllljjjjjjjjjjjjjjjjjjjjjj eeeiiillllljjjjjjjjjjjjjjjjjjjjjj eeeiiillllljjjjjjjjjjjjjjjjjjjjjj eeeiiillllljjjjjjjjjjjjjjjjjjjjjj eeeiiikkkkkkkkkkkkkkkkkkkkkkkkkkk eeeiiikkkkkkkkkkkkkkkkkkkkkkkkkkk eeeiiikkkkkkkkkkkkkkkkkkkkkkkkkkk eeedddddddddddddddddddddddddddddd eeedddddddddddddddddddddddddddddd eeedddddddddddddddddddddddddddddd a(34)=12 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbb ffoooooooooooooooogggggggggggggbbb ffoooooooooooooooogggggggggggggbbb ffoooooooooooooooogggggggggggggbbb ffoooooooooooooooogggggggggggggbbb ffnnnnnnmmmpppppppgggggggggggggbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmppppppprrrrrrrrhhhhhbbb ffnnnnnnmmmqqqqqqqqqqqqqqqhhhhhbbb ffnnnnnnmmmqqqqqqqqqqqqqqqhhhhhbbb ffllllllmmmqqqqqqqqqqqqqqqhhhhhbbb ffllllllmmmqqqqqqqqqqqqqqqhhhhhbbb ffllllllmmmiiiiiiiiiiiiiiiiiiiibbb ffllllllmmmiiiiiiiiiiiiiiiiiiiibbb ffllllllmmmiiiiiiiiiiiiiiiiiiiibbb ffllllllmmmjjjjjjjjjjjjjjddddddddd ffllllllmmmjjjjjjjjjjjjjjddddddddd ffllllllmmmjjjjjjjjjjjjjjddddddddd ffllllllmmmjjjjjjjjjjjjjjddddddddd ffllllllmmmjjjjjjjjjjjjjjddddddddd ffkkkkkkkkkkkkkkkkkkkkkkkddddddddd ffkkkkkkkkkkkkkkkkkkkkkkkddddddddd ffkkkkkkkkkkkkkkkkkkkkkkkddddddddd ffeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee ffeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee cccccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccc a(35)=11 aaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbb jjggggggfffffffffffffffffffffbbbbbb jjggggggfffffffffffffffffffffbbbbbb jjggggggfffffffffffffffffffffbbbbbb jjgggggghhhhhhhhiiiiiiiiiiiiibbbbbb jjgggggghhhhhhhhiiiiiiiiiiiiibbbbbb jjgggggghhhhhhhhiiiiiiiiiiiiibbbbbb jjgggggghhhhhhhhiiiiiiiiiiiiibbbbbb jjgggggghhhhhhhhiiiiiiiiiiiiibbbbbb jjgggggghhhhhhhhddddddddddddddddddd jjgggggghhhhhhhhddddddddddddddddddd jjgggggghhhhhhhhddddddddddddddddddd jjeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee jjeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee jjllllllllllllllllllllllllllllllccc jjllllllllllllllllllllllllllllllccc jjmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnccc jjmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnccc jjmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnccc jjqqqqqqqpppppppppppppppnnnnnnnnccc jjqqqqqqqpppppppppppppppnnnnnnnnccc jjqqqqqqqpppppppppppppppnnnnnnnnccc jjqqqqqqqpppppppppppppppnnnnnnnnccc jjqqqqqqqrrrrrrrrrrrroooooooooooccc jjqqqqqqqrrrrrrrrrrrroooooooooooccc jjqqqqqqqrrrrrrrrrrrroooooooooooccc jjqqqqqqqrrrrrrrrrrrroooooooooooccc jjqqqqqqqrrrrrrrrrrrroooooooooooccc jjssssssssssssssttttttttttttttttccc jjssssssssssssssttttttttttttttttccc jjssssssssssssssttttttttttttttttccc jjssssssssssssssttttttttttttttttccc kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkccc kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkccc a(36)=12 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbb gggggggghhhiimmmfffffffffffffffffbbb gggggggghhhiimmmfffffffffffffffffbbb gggggggghhhiimmmfffffffffffffffffbbb gggggggghhhiimmmfffffffffffffffffbbb gggggggghhhiimmmrrrqqqqqqqqqqqqqqbbb gggggggghhhiimmmrrrqqqqqqqqqqqqqqbbb gggggggghhhiimmmrrrqqqqqqqqqqqqqqbbb gggggggghhhiimmmrrrqqqqqqqqqqqqqqbbb kkklllllhhhiimmmrrrqqqqqqqqqqqqqqbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooppppppbbb kkklllllhhhiimmmrrrssssooooddddddddd kkklllllhhhiimmmrrrssssooooddddddddd kkklllllhhhiimmmrrrssssooooddddddddd kkkjjjjjjjjiimmmrrrssssooooddddddddd kkkjjjjjjjjiimmmrrrssssooooddddddddd kkkjjjjjjjjiinnnnnnnnnnooooddddddddd kkkjjjjjjjjiinnnnnnnnnnooooddddddddd kkkjjjjjjjjiinnnnnnnnnneeeeeeeeeeeee kkkjjjjjjjjiinnnnnnnnnneeeeeeeeeeeee kkkjjjjjjjjiinnnnnnnnnneeeeeeeeeeeee kkkjjjjjjjjiinnnnnnnnnneeeeeeeeeeeee kkkjjjjjjjjiinnnnnnnnnneeeeeeeeeeeee cccccccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccccc a(37)=11 aaaaaaaaaaaaaaaaffffffffffffffffffbbb aaaaaaaaaaaaaaaaffffffffffffffffffbbb aaaaaaaaaaaaaaaaffffffffffffffffffbbb aaaaaaaaaaaaaaaaffffffffffffffffffbbb aaaaaaaaaaaaaaaaffffffffffffffffffbbb llllkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkbbb llllkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkbbb llllkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkbbb llllnnnnnnmmmmmmmmmmmmmmmjjjjjjjjjbbb llllnnnnnnmmmmmmmmmmmmmmmjjjjjjjjjbbb llllnnnnnnmmmmmmmmmmmmmmmjjjjjjjjjbbb llllnnnnnnmmmmmmmmmmmmmmmjjjjjjjjjbbb llllnnnnnnmmmmmmmmmmmmmmmjjjjjjjjjbbb llllnnnnnnmmmmmmmmmmmmmmmjjjjjjjjjbbb llllnnnnnnppppppppppphhhhjjjjjjjjjbbb llllnnnnnnppppppppppphhhhjjjjjjjjjbbb llllnnnnnnppppppppppphhhhjjjjjjjjjbbb llllnnnnnnppppppppppphhhhjjjjjjjjjbbb llllnnnnnnppppppppppphhhhiiiiiiiiibbb llllnnnnnnppppppppppphhhhiiiiiiiiibbb llllnnnnnnppppppppppphhhhiiiiiiiiibbb llllnnnnnnppppppppppphhhhiiiiiiiiibbb llllooooooooooooooooohhhhiiiiiiiiibbb llllooooooooooooooooohhhhiiiiiiiiibbb llllooooooooooooooooohhhhiiiiiiiiibbb llllooooooooooooooooohhhhiiiiiiiiibbb llllooooooooooooooooohhhhiiiiiiiiibbb eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeggggggggggggghhhhdddddddddddd eeeeeeeeccccccccccccccccccccccccccccc eeeeeeeeccccccccccccccccccccccccccccc eeeeeeeeccccccccccccccccccccccccccccc a(38)=10 aaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbb ccccdddddddeeeeeeeeeebbbbbbbbbbbbbbbbb ccccdddddddeeeeeeeeeebbbbbbbbbbbbbbbbb ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddeeeeeeeeeeffffffffffffggggg ccccdddddddiiiiiiiiiiiiiiiiiiiiiiggggg ccccdddddddiiiiiiiiiiiiiiiiiiiiiiggggg ccccdddddddiiiiiiiiiiiiiiiiiiiiiiggggg ccccdddddddiiiiiiiiiiiiiiiiiiiiiiggggg ccccdddddddiiiiiiiiiiiiiiiiiiiiiiggggg ccccdddddddiiiiiiiiiiiiiiiiiiiiiiggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg ccccjjjjjjjjjjjjjjkkkkkkkkkkkkkkkggggg cccchhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh cccchhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh cccchhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh cccchhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh a(39)=11 aaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeebbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeebbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeebbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeebbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeebbbbb ddddhhhhhhhhhhhhhhhhhhhhhhheeeeeeebbbbb ddddhhhhhhhhhhhhhhhhhhhhhhheeeeeeebbbbb ddddhhhhhhhhhhhhhhhhhhhhhhheeeeeeebbbbb ddddhhhhhhhhhhhhhhhhhhhhhhheeeeeeebbbbb ddddhhhhhhhhhhhhhhhhhhhhhhheeeeeeebbbbb ddddhhhhhhhhhhhhhhhhhhhhhhheeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjiiiiiiiiiiiiiiiiieeeeeeebbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkgggggggggggggggbbbbb ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddjjjjjjkkkkkkkkkffffffffffffffffffff ddddccccccccccccccccccccccccccccccccccc ddddccccccccccccccccccccccccccccccccccc ddddccccccccccccccccccccccccccccccccccc ddddccccccccccccccccccccccccccccccccccc a(40)=12 aaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbb dddddddddeeeeeeeeeeeeeeeeeeebbbbbbbbbbbb dddddddddeeeeeeeeeeeeeeeeeeebbbbbbbbbbbb dddddddddeeeeeeeeeeeeeeeeeeebbbbbbbbbbbb dddddddddeeeeeeeeeeeeeeeeeeebbbbbbbbbbbb dddddddddeeeeeeeeeeeeeeeeeeebbbbbbbbbbbb dddddddddeeeeeeeeeeeeeeeeeeebbbbbbbbbbbb dddddddddffffffffffffffffffffgggggghhhhh dddddddddffffffffffffffffffffgggggghhhhh dddddddddffffffffffffffffffffgggggghhhhh dddddddddffffffffffffffffffffgggggghhhhh dddddddddffffffffffffffffffffgggggghhhhh dddddddddffffffffffffffffffffgggggghhhhh lllllllllllllllllllllllllllllgggggghhhhh lllllllllllllllllllllllllllllgggggghhhhh lllllllllllllllllllllllllllllgggggghhhhh lllllllllllllllllllllllllllllgggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh mmmmmmmmmmmmmmnnnnnnnnnnnnnnngggggghhhhh kkkkkkkkkkkkkjjjjjjjjjjjjjjjjjjjjjjhhhhh kkkkkkkkkkkkkjjjjjjjjjjjjjjjjjjjjjjhhhhh kkkkkkkkkkkkkjjjjjjjjjjjjjjjjjjjjjjhhhhh kkkkkkkkkkkkkjjjjjjjjjjjjjjjjjjjjjjhhhhh kkkkkkkkkkkkkjjjjjjjjjjjjjjjjjjjjjjhhhhh kkkkkkkkkkkkkiiiiiiiiiiiiiiiiiiiiiiiiiii kkkkkkkkkkkkkiiiiiiiiiiiiiiiiiiiiiiiiiii kkkkkkkkkkkkkiiiiiiiiiiiiiiiiiiiiiiiiiii kkkkkkkkkkkkkiiiiiiiiiiiiiiiiiiiiiiiiiii cccccccccccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccccccccccc a(41)=13 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaccccbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaccccbbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaccccbbb mmmmmmmmnnnnkkkkkkkkkkkkkkkkkkkkkkccccbbb mmmmmmmmnnnnkkkkkkkkkkkkkkkkkkkkkkccccbbb mmmmmmmmnnnnkkkkkkkkkkkkkkkkkkkkkkccccbbb mmmmmmmmnnnnkkkkkkkkkkkkkkkkkkkkkkccccbbb mmmmmmmmnnnnkkkkkkkkkkkkkkkkkkkkkkccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnlllllllllllllllljjjjjjccccbbb mmmmmmmmnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiijjjjjjccccbbb ppppppppnnnnooooooohhhhiiiiiffffffffffbbb ppppppppnnnnooooooohhhhiiiiiffffffffffbbb ppppppppnnnnooooooohhhhiiiiiffffffffffbbb ppppppppnnnnooooooohhhhiiiiiffffffffffbbb ppppppppnnnnooooooohhhhiiiiiffffffffffbbb eeeeeeeeeeggggggggghhhhiiiiiffffffffffbbb eeeeeeeeeeggggggggghhhhiiiiiffffffffffbbb eeeeeeeeeeggggggggghhhhiiiiiffffffffffbbb eeeeeeeeeeggggggggghhhhiiiiiffffffffffbbb eeeeeeeeeeggggggggghhhhiiiiiffffffffffbbb eeeeeeeeeeggggggggghhhhdddddddddddddddddd eeeeeeeeeeggggggggghhhhdddddddddddddddddd eeeeeeeeeeggggggggghhhhdddddddddddddddddd eeeeeeeeeeggggggggghhhhdddddddddddddddddd eeeeeeeeeeggggggggghhhhdddddddddddddddddd eeeeeeeeeeggggggggghhhhdddddddddddddddddd a(42)=12 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa eeeddddddddddddddddddddddddddddddddddddddd eeeddddddddddddddddddddddddddddddddddddddd eeennnnnnnnnnnnnnnnnnnnnnnnnnnnngggggggggg eeennnnnnnnnnnnnnnnnnnnnnnnnnnnngggggggggg eeennnnnnnnnnnnnnnnnnnnnnnnnnnnngggggggggg eeeqqqqqqqqqqqqqqqoooooooooooooogggggggggg eeeqqqqqqqqqqqqqqqoooooooooooooogggggggggg eeeqqqqqqqqqqqqqqqoooooooooooooogggggggggg eeeqqqqqqqqqqqqqqqoooooooooooooogggggggggg eeeqqqqqqqqqqqqqqqoooooooooooooogggggggggg eeeqqqqqqqqqqqqqqqoooooooooooooollllkkkccc eeerrrrrrrppppppppppppppppppppppllllkkkccc eeerrrrrrrppppppppppppppppppppppllllkkkccc eeerrrrrrrppppppppppppppppppppppllllkkkccc eeerrrrrrrppppppppppppppppppppppllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeerrrrrrruuuuuuuuuuuttttttmmmmmllllkkkccc eeessssssssssssssssssttttttmmmmmllllkkkccc eeessssssssssssssssssttttttmmmmmllllkkkccc eeessssssssssssssssssttttttmmmmmllllkkkccc eeessssssssssssssssssttttttmmmmmllllkkkccc eeessssssssssssssssssttttttmmmmmllllkkkccc fffffffffffffffffffffffffffmmmmmllllkkkccc fffffffffffffffffffffffffffmmmmmllllkkkccc fffffffffffffffffffffffffffmmmmmllllkkkccc bbbbbbbbbbhhhhhhhhhiiiiiiiiiiiiiiiiikkkccc bbbbbbbbbbhhhhhhhhhiiiiiiiiiiiiiiiiikkkccc bbbbbbbbbbhhhhhhhhhiiiiiiiiiiiiiiiiikkkccc bbbbbbbbbbhhhhhhhhhiiiiiiiiiiiiiiiiikkkccc bbbbbbbbbbhhhhhhhhhiiiiiiiiiiiiiiiiikkkccc bbbbbbbbbbhhhhhhhhhjjjjjjjjjjjjjjjjjjjjccc bbbbbbbbbbhhhhhhhhhjjjjjjjjjjjjjjjjjjjjccc bbbbbbbbbbhhhhhhhhhjjjjjjjjjjjjjjjjjjjjccc bbbbbbbbbbhhhhhhhhhjjjjjjjjjjjjjjjjjjjjccc a(43)=12 aaaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeebbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeebbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeebbb aaaaaaaaaaaaaaaaaaaaaaaaaaaaeeeeeeeeeeeebbb fffffffffhhhhhhhhhhhhhhhhhhheeeeeeeeeeeebbb fffffffffhhhhhhhhhhhhhhhhhhheeeeeeeeeeeebbb fffffffffhhhhhhhhhhhhhhhhhhheeeeeeeeeeeebbb fffffffffhhhhhhhhhhhhhhhhhhheeeeeeeeeeeebbb fffffffffhhhhhhhhhhhhhhhhhhheeeeeeeeeeeebbb fffffffffhhhhhhhhhhhhhhhhhhheeeeeeeeeeeebbb fffffffffiiiiiiiiiiiiiiiiiiiijjjjjjjkkkkbbb fffffffffiiiiiiiiiiiiiiiiiiiijjjjjjjkkkkbbb fffffffffiiiiiiiiiiiiiiiiiiiijjjjjjjkkkkbbb fffffffffiiiiiiiiiiiiiiiiiiiijjjjjjjkkkkbbb fffffffffiiiiiiiiiiiiiiiiiiiijjjjjjjkkkkbbb fffffffffiiiiiiiiiiiiiiiiiiiijjjjjjjkkkkbbb gggggggggggggggggggggggggggggjjjjjjjkkkkbbb gggggggggggggggggggggggggggggjjjjjjjkkkkbbb gggggggggggggggggggggggggggggjjjjjjjkkkkbbb gggggggggggggggggggggggggggggjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnmmmmmmmmmmmmmmmmjjjjjjjkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddnnnnnnnnpppppppppppppplllllllllkkkkbbb dddddoooooooooooooooooooooolllllllllkkkkbbb dddddoooooooooooooooooooooolllllllllkkkkbbb dddddoooooooooooooooooooooolllllllllkkkkbbb dddddoooooooooooooooooooooolllllllllkkkkbbb dddddoooooooooooooooooooooolllllllllkkkkbbb dddddcccccccccccccccccccccccccccccccccccccc dddddcccccccccccccccccccccccccccccccccccccc dddddcccccccccccccccccccccccccccccccccccccc a(44)=12 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa ddddffffffffffffffffffffffffffffiiiiiiiijjjj ddddffffffffffffffffffffffffffffiiiiiiiijjjj ddddffffffffffffffffffffffffffffiiiiiiiijjjj ddddffffffffffffffffffffffffffffiiiiiiiijjjj ddddffffffffffffffffffffffffffffiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddgggggggggggggggggkkkkkkkkkkkiiiiiiiijjjj ddddeeeeeelllllllllllkkkkkkkkkkkiiiiiiiijjjj ddddeeeeeelllllllllllkkkkkkkkkkkiiiiiiiijjjj ddddeeeeeelllllllllllkkkkkkkkkkkiiiiiiiijjjj ddddeeeeeelllllllllllkkkkkkkkkkkiiiiiiiijjjj ddddeeeeeelllllllllllkkkkkkkkkkkiiiiiiiijjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeelllllllllllmmmmmmmmmmmmmmmmmmmjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhnnnnnnnnnnnnnnnnjjjj ddddeeeeeehhhhhhhhhhhhhhcccccccccccccccccccc bbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc bbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc bbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc bbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc bbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc bbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc a(47)=12 aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb aaaaaaaaaaaaaaaddddddddddddddeeeeebbbbbbbbbbbbb hhhhiiiiijjjjjjddddddddddddddeeeeebbbbbbbbbbbbb hhhhiiiiijjjjjjooooooppppppppeeeeebbbbbbbbbbbbb hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjooooooppppppppeeeeefffffffffgggg hhhhiiiiijjjjjjoooooonnnnnnnnnnnnnnnnnnnnnngggg hhhhiiiiijjjjjjoooooonnnnnnnnnnnnnnnnnnnnnngggg hhhhiiiiijjjjjjoooooonnnnnnnnnnnnnnnnnnnnnngggg hhhhiiiiijjjjjjoooooonnnnnnnnnnnnnnnnnnnnnngggg hhhhiiiiijjjjjjoooooonnnnnnnnnnnnnnnnnnnnnngggg hhhhiiiiijjjjjjoooooonnnnnnnnnnnnnnnnnnnnnngggg hhhhiiiiillllllllllllllllllllllllllllllllllgggg hhhhiiiiillllllllllllllllllllllllllllllllllgggg hhhhiiiiillllllllllllllllllllllllllllllllllgggg hhhhiiiiillllllllllllllllllllllllllllllllllgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg hhhhkkkkkkkkkkkkkkkkkkkmmmmmmmmmmmmmmmmmmmmgggg ccccccccccccccccccccccccccccccccccccccccccccccc ccccccccccccccccccccccccccccccccccccccccccccccc ccccccccccccccccccccccccccccccccccccccccccccccc 

20161202, 17:29  #21 
Nov 2016
10000_{2} Posts 
The new solutions look great. I'll need to incorporate them.
I made a poster out of solutions to size 32. http://community.wolfram.com/groups//m/t/973362 In talks with Hannes Bassen, we've looked at the defect=0 problem a bit. Smallest candidates are {480, 16}, {840, 21}, {924, 14}, {1008, 14}, {1008, 16}, {1050, 15}. Any nrectangle solution will correspond to a polyhedral graph with n edges, possibly with extra rectangles added to the sides. and those don't explode too quickly. http://www.numericana.com/data/polycount.htm There are 5623571 graphs with 24 edges, for example, and that's well within the bounds of a search. A bit more on Blanche dissections at http://community.wolfram.com/groups//m/t/903043 . With an expanded candidate list, it should be possible to analyze large low defect solutions. Most of the solutions found only have solitary cases where rectangles share a full edge. Splitting each edge in the polyhedral graphs won't expand things too much. For a list of polyhedral graphs and canonical forms, see http://demonstrations.wolfram.com/CanonicalPolyhedra/ . Perhaps some of these can be packed. They are too big for http://burrtools.sourceforge.net/ to handle. {480, 16, 16, {30*480, 32*450, 36*400, 40*360, 45*320, 48*300, 50*288, 60*240, 64*225, 72*200, 75*192, 80*180, 90*160, 96*150, 100*144, 120*120}} {840, 21, 21, {40*840, 42*800, 48*700, 50*672, 56*600, 60*560, 64*525, 70*480, 75*448, 80*420, 84*400, 96*350, 100*336, 105*320, 112*300, 120*280, 140*240, 150*224, 160*210, 168*200, 175*192}} {924, 14, 14, {66*924, 72*847, 77*792, 84*726, 88*693, 99*616, 121*504, 126*484, 132*462, 154*396, 168*363, 198*308, 231*264, 242*252}} {1008, 16, 14, {72*1008, 81*896, 84*864, 96*756, 108*672, 112*648, 126*576, 128*567, 144*504, 162*448, 168*432, 189*384, 192*378, 216*336, 224*324, 252*288}} {1008, 16, 16, {63*1008, 72*882, 81*784, 84*756, 98*648, 108*588, 112*567, 126*504, 144*441, 147*432, 162*392, 168*378, 189*336, 196*324, 216*294, 252*252}} {1050, 15, 15, {70*1050, 75*980, 84*875, 98*750, 100*735, 105*700, 125*588, 140*525, 147*500, 150*490, 175*420, 196*375, 210*350, 245*300, 250*294}} 
20161202, 19:36  #22 
Nov 2016
20_{8} Posts 
So far, we seem to have the sequence
2, 4, 4, 5, 5, 6, 6, 8, 6, 7, 8, 6, 8, 8, 8, 8, 8, 9, 9, 9, 8, 9, 10, 9, 10, 9, 9, 11, 11, 10, 12, 12, 11, 12, 11, 10, 11, 12, 13, 12, 12 I also have decent but nonoptimal solutions up to n=999. Based on the point plot of best known solutions to 999, a good upper bound seems to be n/log(n) + 3. So far, the differences between that upper bound and the known optimal values are: 4, 2, 3, 2, 2, 1, 2, 0, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 1, 2, 2, 3, 3, 1, 2, 3, 1, 1, 2, 2, 3, 4, 3, 2, 2, 3, 3 The next 4 values I have are at 64, 176, 241, 245, 289. I have 5 values at 86 and 139. I also have a 7 value for square 280, defect 46 (71447098) with 11 rectangles. Under 105, the only solutions I have going over the upper bound are for squares 61, 74, 78, 92, and 99. Nonoptimal solutions that match or beat the upper bound would be useful for those squares. Last fiddled with by EdPeggJr on 20161202 at 20:04 
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