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Old 2016-11-30, 22:06   #12
R. Gerbicz
 
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"Robert Gerbicz"
Oct 2005
Hungary

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Quote:
Originally Posted by cuBerBruce View Post
Good job, R. Gerbicz! Wow, I am surprised that as many as 6 cases thought to be proven optimal have been improved. I haven't rigorously checked your solutions, but I've verified with my own solver code that the scores for at least the smaller ones (up to 23x23) are achievable.
Thanks!

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
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Old 2016-12-01, 02:29   #13
LaurV
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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...
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Old 2016-12-01, 03:10   #14
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Quote:
Originally Posted by LaurV View Post
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...
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.
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Old 2016-12-01, 05:12   #15
EdPeggJr
 
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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 non-rational solutions are then rounded to integers 10 to 999, and the improvements are noted. All results are loosely based on valence-3 graphs, but there are plenty of solutions from order-2 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 2016-12-01 at 05:12
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Old 2016-12-01, 21:35   #16
R. Gerbicz
 
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Quote:
Originally Posted by LaurV View Post
Question: when you say is optimal, does it means that you made exhaustive search? It somehow escapes me how this can be done efficiently...
Yes, these has been found with an exhaustive search, hence there is no better solution. Still working on some larger n values.
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Old 2016-12-01, 23:06   #17
science_man_88
 
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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]==c||y[#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
)
I'm guessing I have some major flaw as I'm not getting good results for most values I try. but he's the code I've been making ( made different version since this thread started but this is the one I haven't closed PARI without copying somewhere ( here) yet.
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Old 2016-12-02, 11:12   #18
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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 work-in-progress):
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
a(38)=10
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
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Old 2016-12-02, 15:10   #19
science_man_88
 
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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.
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Old 2016-12-02, 15:40   #20
R. Gerbicz
 
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Quote:
Originally Posted by HannesB View Post
I'd like to add the following two solutions (while others are still work-in-progress):
a(37)=11
a(38)=10
Yeah, those were easy cases, just to fill the gaps:
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
R. Gerbicz is offline   Reply With Quote
Old 2016-12-02, 17:29   #21
EdPeggJr
 
Nov 2016

24 Posts
Default

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 n-rectangle 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}}
EdPeggJr is offline   Reply With Quote
Old 2016-12-02, 19:36   #22
EdPeggJr
 
Nov 2016

24 Posts
Default

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 non-optimal 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 (7144-7098) with 11 rectangles.

Under 105, the only solutions I have going over the upper bound are for squares 61, 74, 78, 92, and 99. Non-optimal solutions that match or beat the upper bound would be useful for those squares.

Last fiddled with by EdPeggJr on 2016-12-02 at 20:04
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