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 2016-12-30, 02:36 #34 EdPeggJr   Nov 2016 24 Posts Thanks for the note. I've updated a(101). Any chance you can beat my a(99) = 27? That's my only value under 100 that goes past my upper bound. Another that might be easy to beat is a(60)=18.
2016-12-30, 23:55   #35
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

101100010112 Posts

Quote:
 Originally Posted by EdPeggJr Any chance you can beat my a(99) = 27? That's my only value under 100 that goes past my upper bound. Another that might be easy to beat is a(60)=18.
Found nothing for a(99), but for a(60):

Code:
a(60)<=16
aaaaaaaaaaaaaaaagggggffffffffffffffffffffffffffffffffffbbbbb
aaaaaaaaaaaaaaaagggggffffffffffffffffffffffffffffffffffbbbbb
aaaaaaaaaaaaaaaagggggffffffffffffffffffffffffffffffffffbbbbb
aaaaaaaaaaaaaaaagggggffffffffffffffffffffffffffffffffffbbbbb
aaaaaaaaaaaaaaaagggggffffffffffffffffffffffffffffffffffbbbbb
aaaaaaaaaaaaaaaagggggffffffffffffffffffffffffffffffffffbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
aaaaaaaaaaaaaaaagggggooooooooooooooooooooppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggooooooooooooooooooooppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggooooooooooooooooooooppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggooooooooooooooooooooppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggooooooooooooooooooooppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnqqqqqqqqqqqqqqppppppppjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
eeeeeeeeehhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnkkkkkkkkkkkkkkkkkkkkkkjjjjjjbbbbb
dddddddddhhhhhhhgggggnnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiinnnnnnmmmmmmmmmmmmmmmmmmmmmcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc
dddddddddiiiiiiiiiiiilllllllllllllllllllllllllllcccccccccccc

 2017-01-01, 13:42 #36 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 101100010112 Posts For the modified puzzle (when different orientation of the rectangle is permitted) computed up to n=45, see the tilings at https://oeis.org/A279596/a279596.txt One crazy tiling is for n=29, this is using 34 rectangles: Code: a(29)=7 aaaaaaaaaaaaaaaaaaaaaaaaaaaeb dhhhhhhhhhhhhhhhhhhhhhhhhhheb dgkkkkkkkkkkkkkkkkkkkkkkkkkeb dgjnnnnnnnnnnnnnnnnnnnnnnnneb dgjmmooooooooooooooooooooooeb dgjmmiqBBBBBHHHHHHHHHHHHEEEeb dgjmmiqBBBBBHHHHHHHHHHHHEEEeb dgjmmiqBBBBBGGGGGGGGFFFFEEEeb dgjmmiqBBBBBGGGGGGGGFFFFEEEeb dgjmmiqBBBBBGGGGGGGGFFFFEEEeb dgjmmiqCCCCCCCCCCCCCFFFFEEEeb dgjmmiqCCCCCCCCCCCCCFFFFEEEeb dgjmmiqttDDDDDDDDDDDFFFFEEEeb dgjmmiqttDDDDDDDDDDDFFFFEEEeb dgjmmiqttvvzzzzzzzAAAAAAAAAeb dgjmmiqttvvzzzzzzzAAAAAAAAAeb dgjmmiqttvvzzzzzzzAAAAAAAAAeb dgjlliqttvvzzzzzzzwwwyyyyyyeb dgjlliqttvvuuuxxxxwwwyyyyyyeb dgjlliqttvvuuuxxxxwwwyyyyyyeb dgjlliqttvvuuuxxxxwwwyyyyyyeb dgjlliqttvvuuuxxxxwwwrrrrrrrb dgjlliqttvvuuuxxxxwwwrrrrrrrb dgjlliqttvvuuuxxxxwwwrrrrrrrb dgjlliqttvvuuussssssssssssssb dgjlliqttvvuuussssssssssssssb dgjlliqpppppppppppppppppppppb dgjllifffffffffffffffffffffff dcccccccccccccccccccccccccccc
 2017-01-02, 18:05 #37 EdPeggJr   Nov 2016 24 Posts Very interesting solutions. This time, I already managed to find some of those optimal solutions already, but it's good to have all of them. Which one of those required the most time to find?
2017-01-02, 19:34   #38
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

58B16 Posts

Quote:
 Originally Posted by EdPeggJr Which one of those required the most time to find?
The last one: n=45 was the hardest, it took roughly 1 day (do not have the precise time).

 2017-01-04, 16:23 #39 EdPeggJr   Nov 2016 1610 Posts If it would be easy, for values up to 24 or so (or whatever is easy), could you post solutions optimized for the most rectangles instead of the least rectangles? For example, in the Variant for square 31, I have optimal solutions with 19 rectangles and 36 rectangles. Is that the full range? What other numbers of rectangles offer the optimal solution for square 31? I'd like to make a puzzle called the Mondrian Cube, where a square with many solutions has lengths on the four edges so that a cube can be folded.
2017-01-05, 09:06   #40
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

101100010112 Posts

Quote:
 Originally Posted by EdPeggJr If it would be easy, for values up to 24 or so (or whatever is easy), could you post solutions optimized for the most rectangles instead of the least rectangles?
After a half day of search the possible number of rectangles in optimal tilings for n=3..29

ps. Note that I've used a slightly different strategy: in the main search for fixed defect looking for tiling in descending order of the minimal rectangle's area from those rectangles that are in the square's corner. [The tiling is started with that rectangle with letter 'a', in the square's upper left corner]. See for example in the oeis sequence for n=5 the example used 7 rectangles, but there is also a tiling for 6 rectangles.
Code:
a(3)=2; 3 rectangles;
aab
ccb
ccb

a(4)=2; 5 rectangles;
aaab
eedb
eedb
cccc

a(5)=3; 6 rectangles;
aaaab
edddb
efffb
efffb
ccccc

a(5)=3; 7 rectangles;
aaaab
fggdb
fggdb
feeeb
ccccc

a(6)=3; 8 rectangles;
aaaaab
efdddb
efhhhb
efhhhb
eggggb
cccccc

a(7)=3; 10 rectangles;
aaaaabb
djihhbb
djihhbb
djifffe
djifffe
djgggge
dcccccc

a(8)=3; 9 rectangles;
aaaaaacb
giieeecb
giieeecb
giifffcb
giifffcb
ghhfffcb
ghhddddb
ghhddddb

a(9)=3; 11 rectangles;
aaaaaaaab
eeddddddb
eeiijjjgb
eeiijjjgb
eeiijjjgb
hhhkkkkgb
hhhkkkkgb
fffffffgb
ccccccccc

a(10)=4; 10 rectangles;
aaaaaaaaab
dddhhhhjjb
dddhhhhjjb
dddggiijjb
eeeggiijjb
eeeggiijjb
eeeggiijjb
eeeggffffb
cccccffffb
cccccffffb

a(10)=4; 12 rectangles;
aaaaaaaaab
effddddddb
effkkklljb
effkkklljb
effkkklljb
ehhhhhlljb
ehhhhhlljb
eiiiiiiijb
eggggggggb
cccccccccc

a(10)=4; 13 rectangles;
aaaaaaaaab
effddddddb
effiijjjhb
effiijjjhb
effiikkkhb
ellllkkkhb
ellllkkkhb
emmmmmmmhb
eggggggggb
cccccccccc

a(10)=4; 14 rectangles;
aaaaaacbbb
fnmljjcbbb
fnmljjcddd
fnmljjcddd
fnmljjcddd
fnmliiiigg
fnmliiiigg
fnmkkkkkgg
fnhhhhhhhh
feeeeeeeee

a(11)=4; 12 rectangles;
aaaaaaaaaab
eeddddddddb
eekkkklllgb
eekkkklllgb
eekkkklllgb
eejjjjlllgb
eejjjjiiigb
hhhhhhiiigb
hhhhhhiiigb
fffffffffgb
ccccccccccc

a(11)=4; 15 rectangles;
aaaaaaaaaeb
diiiiiiiieb
dhllllllleb
dhkoooonneb
dhkoooonneb
dhkjjmmnneb
dhkjjmmgggb
dhkjjmmgggb
dhkjjmmgggb
dhkjjffffff
dcccccccccc

a(12)=4; 14 rectangles;
aaaaaaaaaaab
effddddddddb
effmmmmnnnjb
effmmmmnnnjb
effmmmmnnnjb
effllllnnnjb
effllllkkkjb
ehhhhhhkkkjb
ehhhhhhkkkjb
eiiiiiiiiijb
eggggggggggb
cccccccccccc

a(13)=4; 16 rectangles;
aaaaaaaaaaaeb
diiiiiimmmleb
diiiiiimmmleb
djjjjnnmmmleb
djjjjnnpppleb
djjjjnnpppleb
dhhkknnpppleb
dhhkknnpppleb
dhhkkoooooleb
dhhkkoooooleb
dhhgggggggggb
dhhffffffffff
dcccccccccccc

a(13)=4; 17 rectangles;
aaaaaaaaaaghb
emmmppppqqghb
emmmppppqqghb
emmmppppqqghb
efnnnnnnqqghb
efnnnnnnqqghb
efjjllooooghb
efjjllooooghb
efjjllkkkkkhb
efjjllkkkkkhb
efjjiiiiiiiib
efjjddddddddd
efccccccccccc

a(14)=4; 9 rectangles;
aaaaaaaeeeffbb
aaaaaaaeeeffbb
aaaaaaaeeeffbb
ddiiiiieeeffbb
ddiiiiieeeffbb
ddiiiiieeeffbb
ddiiiiieeeffbb
ddhhhhhhhhffbb
ddhhhhhhhhffbb
ddhhhhhhhhffbb
ddggggggggggbb
ddggggggggggbb
ddcccccccccccc
ddcccccccccccc

a(15)=5; 17 rectangles;
aaaaaaaaaaaaaab
dfffffffffffffb
diiiiiiiiiiiieb
dnnnnnnnnnnnheb
dmmqqqqqqpppheb
dmmqqqqqqpppheb
dmmlloooopppheb
dmmlloooopppheb
dmmlloooopppheb
dmmllooookkkheb
dmmlljjjjkkkheb
dmmlljjjjkkkheb
cccccjjjjkkkheb
cccccgggggggheb
cccccgggggggheb

a(15)=5; 18 rectangles;
aaaaaaaaaaaaaab
eghfddddddddddb
eghfmmmmmlllllb
eghfmmmmmlllllb
eghfmmmmmooonnb
eghfqqrrrooonnb
eghfqqrrrooonnb
eghfqqrrrooonnb
eghfqqrrrooonnb
eghfqqppppppnnb
eghfqqppppppnnb
eghkkkkkkkkkkkb
egjjjjjjjjjjjjb
eiiiiiiiiiiiiib
ccccccccccccccc

a(15)=5; 19 rectangles;
aaaaaaaaaaaaaeb
djjjjjjjjjjjkeb
drrrrrrrrrrmkeb
dpppsssssssmkeb
dpppsssssssmkeb
dpppqqqqqnnmkeb
dpppqqqqqnnmkeb
dhhiioooonnmkeb
dhhiioooonnmkeb
dhhiioooonnmkeb
dhhiillllllmkeb
dhhiillllllmkeb
dhhiigggggggggb
dhhffffffffffff
dcccccccccccccc

a(16)=5; 19 rectangles;
aaaaaaaaaaaaaaab
efffdddddddddddb
efffhhhhhhhhjigb
efffhhhhhhhhjigb
efffmmrrrrrrjigb
ennnmmrrrrrrjigb
ennnmmssssqqjigb
ennnmmssssqqjigb
ennnmmssssqqjigb
ennnmmssssqqjigb
eoooopppppqqjigb
eoooopppppqqjigb
eoooopppppqqjigb
elllllllllllligb
ekkkkkkkkkkkkkgb
cccccccccccccccc

a(16)=5; 20 rectangles;
aaaaaaaaaaaabbbb
dfhigeeeeeeebbbb
dfhigeeeeeeebbbb
dfhignnnnnooooom
dfhignnnnnooooom
dfhigrrqqqooooom
dfhigrrqqqsssppm
dfhigrrqqqsssppm
dfhigrrqqqsssppm
dfhigrrqqqsssppm
dfhigrrttttttppm
dfhigrrttttttppm
dfhilllllllllllm
dfhkkkkkkkkkkkkk
dfjjjjjjjjjjjjjj
dccccccccccccccc

a(17)=5; 7 rectangles;
aaaaaaaaaaaaabbbb
aaaaaaaaaaaaabbbb
aaaaaaaaaaaaabbbb
dddffffggggggbbbb
dddffffggggggbbbb
dddffffggggggbbbb
dddffffggggggbbbb
dddffffggggggbbbb
dddffffggggggbbbb
dddffffggggggbbbb
dddffffeeeeeeeeee
dddffffeeeeeeeeee
dddffffeeeeeeeeee
dddffffeeeeeeeeee
dddcccccccccccccc
dddcccccccccccccc
dddcccccccccccccc

a(17)=5; 21 rectangles;
aaaaaaaaaaaaaaaeb
djjjjjjjjjjjjjjeb
dinnnnnnnnnnnnneb
dimmkoqqqqrrrrreb
dimmkoqqqqrrrrreb
dimmkoqqqqrrrrreb
dimmkopptttuuuueb
dimmkopptttuuuueb
dimmkopptttuuuueb
dimmkopptttuuuueb
dillkoppssssssseb
dillkoppssssssseb
dillkopphhhhhhhhb
dillkopphhhhhhhhb
dillkgggggggggggb
dillkffffffffffff
dcccccccccccccccc

a(18)=6; 12 rectangles;
aaaaaaaaaccbbbbbbb
aaaaaaaaaccbbbbbbb
aaaaaaaaaccbbbbbbb
hhllllkkkccbbbbbbb
hhllllkkkcceeeffdd
hhllllkkkcceeeffdd
hhllllkkkcceeeffdd
hhllllkkkcceeeffdd
hhllllkkkcceeeffdd
hhllllkkkcceeeffdd
hhjjjjkkkcceeeffdd
hhjjjjkkkcceeeffdd
hhjjjjkkkcceeeffdd
hhjjjjiiiiiiiiffdd
hhjjjjiiiiiiiiffdd
hhjjjjiiiiiiiiffdd
hhggggggggggggggdd
hhggggggggggggggdd

a(18)=6; 15 rectangles;
aaaaaaaaaabbbbbbbb
aaaaaaaaaabbbbbbbb
nnnmmooeeebbbbbbbb
nnnmmooeeejjjjggff
nnnmmooeeejjjjggff
nnnmmooeeejjjjggff
nnnmmooeeejjjjggff
nnnmmooeeejjjjggff
nnnmmooeeejjjjggff
nnnmmookkkkkkkggff
lllmmookkkkkkkggff
lllmmookkkkkkkggff
lllmmhhhhhhhhhhhff
lllmmhhhhhhhhhhhff
llliiiiiiiiicccccc
llliiiiiiiiicccccc
ddddddddddddcccccc
ddddddddddddcccccc

a(18)=6; 18 rectangles;
aaaaaaaaaaaaaaaaab
eddddddddddddddddb
eqqqppppppppppgggb
eqqqppppppppppgggb
eqqqrrrrrroooogggb
eqqqrrrrrroooogggb
eqqqrrrrrroooogggb
eiinnnnnnnoooogggb
eiinnnnnnnmmmmmffb
eiinnnnnnnmmmmmffb
eiijjjllllmmmmmffb
eiijjjllllmmmmmffb
eiijjjllllkkkkkffb
eiijjjllllkkkkkffb
eiijjjllllkkkkkffb
eiijjjhhhhhhhhhffb
eiijjjhhhhhhhhhffb
cccccccccccccccccc

a(18)=6; 19 rectangles;
aaaaaaaaaaaaaaaaab
eggddddddddddddddb
eggmmmmmmmmmsssrrb
eggmmmmmmmmmsssrrb
eggnnnppppppsssrrb
eggnnnppppppsssrrb
eggnnnppppppsssrrb
eggnnnoooooqqqqrrb
eggnnnoooooqqqqrrb
eggnnnoooooqqqqrrb
eiiiiiiiiiiqqqqrrb
eiiiiiiiiiiqqqqrrb
ejjjjjjjjjjjjjjjjb
ehhhhhhhlllllkkkkb
ehhhhhhhlllllkkkkb
fffffffflllllkkkkb
fffffffflllllkkkkb
cccccccccccccccccc

a(18)=6; 20 rectangles;
aaaaaaaaaaaaaaaaab
egfddddddddddddddb
egfkkkkklljjjjjjjb
egfkkkkklljjjjjjjb
egfkkkkkllttttsssb
egfnnmmmllttttsssb
egfnnmmmllttttsssb
egfnnmmmllttttsssb
egfnnmmmllttttsssb
egfnnmmmqqqqqqqqqb
egfnnmmmqqqqqqqqqb
egfnnpppppppprrrrb
egfnnpppppppprrrrb
egfoooooooooorrrrb
egfoooooooooorrrrb
egiiiiiiiiiiiiiiib
ehhhhhhhhhhhhhhhhb
cccccccccccccccccc

a(18)=6; 21 rectangles;
aaaaaaaaaaaaaaaaab
ejjjjddddddddddddb
ejjjjmmmmmmmoonngb
ejjjjmmmmmmmoonngb
elllllqqqqqqoonngb
elllllqqqqqqoonngb
elllllqqqqqqoonngb
ekkrrrrrrrrroonngb
ekkrrrrrrrrroonngb
ekktttppppppppnngb
ekktttppppppppnngb
ekktttuuuusssiiigb
ekktttuuuusssiiigb
ekktttuuuusssiiigb
ekktttuuuusssiiigb
hhhhhhhhhhhhhiiigb
ffffffffffffffffgb
cccccccccccccccccc

a(18)=6; 22 rectangles;
aaaaaaaaaaaaaaaaaa
kkkkkkkkkkkkkkkkml
qqqqqqqqqqqqqqrrml
ppppppppppppssrrml
ffffffffdottssrrml
ffffffffdottssrrml
eeeeegggdottssrrml
eeeeegggdottssrrml
eeeeegggdottssrrml
cchhhgggdottvvvvml
cchhhjjjdottvvvvml
cchhhjjjdottvvvvml
cchhhjjjdouuuuuuml
cchhhjjjdouuuuuuml
cchhhjjjdonnnnnnnl
cciiiiiidonnnnnnnl
cciiiiiidbbbbbbbbb
cciiiiiidbbbbbbbbb

a(18)=6; 23 rectangles;
aaaaaaaaaaaaaaaaeb
dhhhhhhhhhhhhhhheb
dgkkkkkkmmmmmmmmeb
dgkkkkkkmmmmmmmmeb
dgjllllllllllllleb
dgjippppppppppppeb
dgjiqqqqqqqttssseb
dgjiqqqqqqqttssseb
dgjioovvwwwttssseb
dgjioovvwwwttssseb
dgjioovvwwwttssseb
dgjioovvwwwttrrrrb
dgjioovvuuuuurrrrb
dgjioovvuuuuurrrrb
dgjioovvuuuuurrrrb
dgjioonnnnnnnnnnnb
dgjiffffffffffffff
dccccccccccccccccc

a(18)=6; 24 rectangles;
aaaaaaaaaaaaaaaaeb
jjjjjjjjjjjjjjjkeb
mmmmmmmmmmmmmmnkeb
lllllllllllllonkeb
iiqqqqqqqqqqqonkeb
iippppppwwwwwonkeb
iippppppwwwwwonkeb
iihsstttwwwwwonkeb
iihsstttxxxvvonkeb
iihsstttxxxvvonkeb
ddhsstttxxxvvonkeb
ddhsstttxxxvvonkeb
ddhrrrruuuuvvonkeb
ddhrrrruuuuvvonkeb
ddhrrrruuuuvvonkeb
ddhrrrrffffffffffb
ddhgggggggcccccccc
ddhgggggggcccccccc

a(18)=6; 25 rectangles;
aaaaaaaaaaaaaaaghb
eeeeeeeeeeuuuuughb
dppposssssuuuuughb
dppposssssuuuuughb
dppporyyyyyyyvvghb
dppporyyyyyyyvvghb
dfqqorxxxxwwwvvghb
dfqqorxxxxwwwvvghb
dfqqorxxxxwwwvvghb
dfqqorxxxxwwwvvghb
dfqqorttttwwwvvghb
dfqqorttttmmmmmmhb
dfnnorttttmmmmmmhb
dfnnorlllllllllllb
dfnnokkkkkkkkkkkkb
dfnnoiiiiiiiiiiiii
dfnnjjjjjjjjjjjjjj
dfcccccccccccccccc

a(19)=6; 11 rectangles;
aaaaaaaaaaaaaaaabbb
aaaaaaaaaaaaaaaabbb
ddfffeeeeeeeeeeebbb
ddfffeeeeeeeeeeebbb
ddfffeeeeeeeeeeebbb
ddfffkkkkkkhhhhhbbb
ddfffkkkkkkhhhhhbbb
ddfffkkkkkkhhhhhbbb
ddfffkkkkkkhhhhhbbb
ddfffkkkkkkhhhhhbbb
ddfffkkkkkkhhhhhbbb
ddfffjjjjjjjjjjcccc
ddfffjjjjjjjjjjcccc
ddfffjjjjjjjjjjcccc
ddggggggiiiiiiicccc
ddggggggiiiiiiicccc
ddggggggiiiiiiicccc
ddggggggiiiiiiicccc
ddggggggiiiiiiicccc

a(19)=6; 17 rectangles;
aaaaaaaaaaaaaaaaaaa
dddddddddhhhhhhhhhh
dddddddddhhhhhhhhhh
eeeeemmmmmmmmjjjjjj
eeeeemmmmmmmmjjjjjj
eeeeemmmmmmmmjjjjjj
eeeeennnnnnnnnnnnii
ccgggnnnnnnnnnnnnii
ccggglllkkqqqqqqqii
ccggglllkkqqqqqqqii
ccggglllkkqqqqqqqii
ccggglllkkppppoooii
ccggglllkkppppoooii
ccggglllkkppppoooii
ccggglllkkppppoooii
ccffffffkkppppoooii
ccffffffkkppppoooii
ccffffffbbbbbbbbbbb
ccffffffbbbbbbbbbbb

a(19)=6; 20 rectangles;
aaaaaaaaaaaaaaaaaab
efddddddddddddddddb
efmmnnnkkkkkkkkkhhb
efmmnnnkkkkkkkkkhhb
efmmnnnoooolllllhhb
efmmnnnoooolllllhhb
efmmnnnoooolllllhhb
efmmnnnoooolllllhhb
efmmnnnoooojjjjjhhb
efmmpppppppjjjjjhhb
efmmpppppppjjjjjhhb
efmmpppppppqqqqiiib
efsssttttttqqqqiiib
efsssttttttqqqqiiib
efsssttttttqqqqiiib
efsssrrrrrrrrrriiib
efsssrrrrrrrrrriiib
egggggggggggggggggb
ccccccccccccccccccc

a(19)=6; 21 rectangles;
aaaaaaaaaaaaaaaaaab
egffddddddddddddddb
egffjjjsssssssssslb
egffjjjsssssssssslb
egffjjjrrqqttttttlb
egffjjjrrqqttttttlb
egffjjjrrqqttttttlb
egffjjjrrqquuuuoolb
egiiiiirrqquuuuoolb
egiiiiirrqquuuuoolb
egiiiiirrqquuuuoolb
egiiiiirrqquuuuoolb
egmmmmmmmqqppppoolb
egmmmmmmmqqppppoolb
egnnnnnnnnnppppoolb
egnnnnnnnnnppppoolb
egkkkkkkkkkkkkkkklb
ehhhhhhhhhhhhhhhhhb
ccccccccccccccccccc

a(19)=6; 22 rectangles;
aaaaaaaaaaaaaaaaaab
eghfddddddddddddddb
eghfmmmmmnnlllllllb
eghfmmmmmnnlllllllb
eghfmmmmmnnvvvvuuub
eghfppooonnvvvvuuub
eghfppooonnvvvvuuub
eghfppooonnvvvvuuub
eghfppooonnvvvvuuub
eghfppooosssssssssb
eghfppooosssssssssb
eghfpprrrrrrrrttttb
eghfpprrrrrrrrttttb
eghfqqqqqqqqqqttttb
eghfqqqqqqqqqqttttb
eghkkkkkkkkkkkkkkkb
egjjjjjjjjjjjjjjjjb
eiiiiiiiiiiiiiiiiib
ccccccccccccccccccc

a(19)=6; 23 rectangles;
aaaaaaaaaaaaaaaaaeb
djjjjjjjjjjjjjjjjeb
dinnnnnnnnnnnnnnneb
dilooooooooooooooeb
dilmuuuttttttttrreb
dilmuuuttttttttrreb
dilmuuuwwwwwwssrreb
dilmuuuwwwwwwssrreb
dilmuuuwwwwwwssrreb
dilmkkvvvvvvvssrreb
dilmkkvvvvvvvssrreb
dilmkkpppqqqqssrreb
dilmkkpppqqqqhhhhhb
dilmkkpppqqqqhhhhhb
dilmkkpppqqqqhhhhhb
dilmkkpppgggggggggb
dilmkkpppgggggggggb
dilmkkfffffffffffff
dcccccccccccccccccc

a(19)=6; 24 rectangles;
aaaaaaaaaaaaaaaaaeb
djjjjjjjoooooooooeb
djjjjjjjoooooooooeb
dinnnnnnnnnnnnnnneb
dilppppppppppppppeb
dilmuuuuuuuuuuuuueb
dilmxxvvvvvvssttteb
dilmxxvvvvvvssttteb
dilmxxvvvvvvssttteb
dilmkkkqqqrrggggggb
dilmkkkqqqrrggggggb
dilmkkkffffffffffff
dcccccccccccccccccc

a(20)=4; 8 rectangles;
aaaaaaaaaaaaaaaabbbb
aaaaaaaaaaaaaaaabbbb
aaaaaaaaaaaaaaaabbbb
dddgggggggggggggbbbb
dddgggggggggggggbbbb
dddgggggggggggggbbbb
dddgggggggggggggbbbb
dddfffffhhhhhhhhbbbb
dddfffffhhhhhhhhbbbb
dddfffffhhhhhhhhbbbb
dddfffffhhhhhhhhbbbb
dddfffffhhhhhhhhbbbb
dddfffffhhhhhhhhbbbb
dddfffffeeeeeeeeeeee
dddfffffeeeeeeeeeeee
dddfffffeeeeeeeeeeee
dddfffffeeeeeeeeeeee
dddccccccccccccccccc
dddccccccccccccccccc
dddccccccccccccccccc

a(21)=6; 13 rectangles;
aaaaaaaaaaaaaaaabbbbb
aaaaaaaaaaaaaaaabbbbb
eeeffffffiiiiiiibbbbb
eeeffffffiiiiiiibbbbb
eeeffffffiiiiiiibbbbb
eeeffffffiiiiiiibbbbb
eeeffffffiiiiiiibbbbb
eeehhhhhhhhhhhhhhhhhh
eeehhhhhhhhhhhhhhhhhh
eeeggggggggggggjjjjjj
eeeggggggggggggjjjjjj
eeeggggggggggggjjjjjj
ddddmmmmmmmmmmmjjjjjj
ddddmmmmmmmmmmmjjjjjj
ddddmmmmmmmmmmmjjjjjj
ddddlllllllllkkkkkkkk
ddddlllllllllkkkkkkkk
ddddlllllllllkkkkkkkk
ddddlllllllllkkkkkkkk
ddddccccccccccccccccc
ddddccccccccccccccccc

a(21)=6; 16 rectangles;
aaaaaaaaaaaaaabbbbbbb
aaaaaaaaaaaaaabbbbbbb
eeffffffffffffbbbbbbb
eeffffffffffffbbbbbbb
eehhhhhhhhhhhhhjjjjii
eehhhhhhhhhhhhhjjjjii
eegggmmmmnnnnnnjjjjii
eegggmmmmnnnnnnjjjjii
eegggmmmmnnnnnnjjjjii
eegggmmmmnnnnnnjjjjii
eegggmmmmppppppppppii
eegggmmmmppppppppppii
eegggmmmmppppppppppii
eegggkkkkkkkkkoooooii
eegggkkkkkkkkkoooooii
eegggkkkkkkkkkoooooii
ddddddlllllllloooooii
ddddddlllllllloooooii
ddddddlllllllloooooii
ddddddccccccccccccccc
ddddddccccccccccccccc

a(21)=6; 21 rectangles;
aaaaaaaaaaaaaaaaaaaaa
ggggggggggggggggggggh
iiiiiioooooooooooopph
iiiiiioooooooooooopph
iiiiiinnnnnnnnnnrrpph
iiiiiinnnnnnnnnnrrpph
ffjjjkkkuuuuuuuurrpph
ffjjjkkkuuuuuuuurrpph
ffjjjkkkuuuuuuuurrpph
ffjjjkkkttttttssrrpph
ffjjjkkkttttttssrrpph
ffjjjkkkttttttssrrpph
ffjjjlllllllllssrrpph
ffjjjlllllllllssqqqqh
ffeeeeedddmmmmssqqqqh
ffeeeeedddmmmmssqqqqh
ffeeeeedddmmmmssqqqqh
ffeeeeedddmmmmssqqqqh
cccccccdddmmmmssqqqqh
cccccccdddbbbbbbbbbbb
cccccccdddbbbbbbbbbbb

a(21)=6; 24 rectangles;
aaaaaaaaaaaaaaaaaaaab
eghfddddddddddddddddb
eghfqqrrrooooooooollb
eghfqqrrrooooooooollb
eghfqqtttttttnnnnnllb
eghfqqtttttttnnnnnllb
eghfqqtttttttuuuummmb
eghfwwwxxxxxxuuuummmb
eghfwwwxxxxxxuuuummmb
eghfwwwxxxxxxuuuummmb
eghfwwwvvvvvvvvvvmmmb
eghfwwwvvvvvvvvvvmmmb
eghkkkkkkkkkkkkkkkkkb
egjjjjjjjjjjjjjjjjjjb
eiiiiiiiiiiiiiiiiiiib
ccccccccccccccccccccc

a(21)=6; 25 rectangles;
aaaaaaaaaaaaaaaaaafhb
eeeeeeeeeeeeeeeeegfhb
dmmooooooooxxxyyrgfhb
dmmooooooooxxxyyrgfhb
dmmnnvvvvvvxxxyyrgfhb
dmmnnvvvvvvxxxyyrgfhb
dmmnnvvvvvvxxxyyrgfhb
dmmnnuuuuwwwwwyyrgfhb
dmmnnuuuuwwwwwyyrgfhb
dmmnnuuuuwwwwwyyrgfhb
dmmnnuuuuwwwwwyyrgfhb
dmmnnuuuuttttsssrgfhb
dlllqqqqqttttsssrgfhb
dlllqqqqqttttsssrgfhb
dlllqqqqqttttsssrgfhb
dlllpppppppppsssrgfhb
dlllpppppppppsssrgfhb
dlllkkkkkkkkkkkkkkkhb
dllljjjjjjjjjjjjjjjjb
diiiiiiiiiiiiiiiiiiib
dcccccccccccccccccccc

a(21)=6; 26 rectangles;
aaaaaaaaaaaaaaaaaaabb
diiiiiiiiiiiiiiiiiibb
dvvvvvvvvvvvvvvvvvhbb
dzxxxxxxxxxxxxxxxxhbb
dzywwwwwwwwwwwwwwwhbb
dzytuuuuuuuuuuuuuuhbb
dzytqqqqqrrrrrrrrrhbb
dzytqqqqqrrrrrrrrrhbb
dzytqqqqqlllllkkkkhbb
dzytqqqqqlllllkkkkhbb
dzytssppplllllkkkkhee
dzytsspppoooggkkkkhee
dzytsspppoooggkkkkhee
dzytsspppoooggjjjjhee
dzytsspppoooggjjjjhee
dzytsspppoooggjjjjhee
dzytssmmmmmmggjjjjhee
dzytssmmmmmmggfffffff
dzytssmmmmmmggfffffff
dzynnnnnnnncccccccccc
dzynnnnnnnncccccccccc

a(22)=6; 13 rectangles;
aaaaaaaaaaaaaaaaabbbbb
aaaaaaaaaaaaaaaaabbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggjjjiiiiddddddddddddd
ggjjjiiiiddddddddddddd
ggjjjiiiiddddddddddddd
ggjjjiiiimmmmmmllllccc
ggjjjiiiimmmmmmllllccc
ggjjjiiiimmmmmmllllccc
ggjjjiiiimmmmmmllllccc
ggjjjiiiimmmmmmllllccc
ggjjjiiiimmmmmmllllccc
ggjjjkkkkkkkkkkllllccc
ggjjjkkkkkkkkkkllllccc
ggjjjkkkkkkkkkkllllccc
ggjjjkkkkkkkkkkllllccc
hhhhhhhhhhhhhhhhhhhccc
hhhhhhhhhhhhhhhhhhhccc

a(22)=6; 18 rectangles;
aaaaaaaaaaaabbbbbbbbbb
aaaaaaaaaaaabbbbbbbbbb
ddddeeccccccbbbbbbbbbb
ddddeeccccccnnnnnnnnll
ddddeeccccccnnnnnnnnll
ddddeeccccccnnnnnnnnll
ddddeeooooooooooooooll
ddddeeooooooooooooooll
ggffeerrrrrrqqqqqmmmll
ggffeerrrrrrqqqqqmmmll
ggffeerrrrrrqqqqqmmmll
ggffeerrrrrrqqqqqmmmll
ggffeerrrrrrqqqqqmmmll
ggffeepppppppkkkkmmmll
ggffeepppppppkkkkmmmll
ggffeepppppppkkkkmmmll
ggffeepppppppkkkkjjjjj
ggffiiiiiiiiikkkkjjjjj
ggffiiiiiiiiikkkkjjjjj
ggffiiiiiiiiikkkkjjjjj
gghhhhhhhhhhhhhhhjjjjj
gghhhhhhhhhhhhhhhjjjjj

a(22)=6; 23 rectangles;
aaaaaaaaaaaaaaaaaaaaab
efdddddddddddddddddddb
efmmmmmmmmmmjjjjjjjjjb
efmmmmmmmmmmjjjjjjjjjb
efpppppppplllllllllllb
efpppppppplllllllllllb
efppppppppvvvvuukkkkkb
efoorrrrrrvvvvuukkkkkb
efoorrrrrrvvvvuukkkkkb
efoorrrrrrvvvvuukkkkkb
efoorrrrrrvvvvuutttiib
efooqqqssswwwwuutttiib
efooqqqssswwwwuutttiib
efooqqqssswwwwuutttiib
efooqqqssswwwwuutttiib
efooqqqssswwwwuutttiib
efooqqqssswwwwuutttiib
efooqqqnnnnnnnnnnnniib
efooqqqnnnnnnnnnnnniib
ehhhhhhhhhhhhhhhhhhiib
eggggggggggggggggggggb
cccccccccccccccccccccc

a(22)=6; 27 rectangles;
ellllllllllllllllllldb
emmmmmppppppqqqqqqqrdb
emmmmmppppppqqqqqqqrdb
emmmmmppppppqqqqqqqrdb
einnnnnnnnnnnnnnnnnrdb
eiooooooooooooooossrdb
eijuuuuuxxxAAAAyyssrdb
eijuuuuuxxxAAAAyyssrdb
eijuuuuuxxxAAAAyyssrdb
eijuuuuuxxxAAAAyyssrdb
eijkkvvvxxxAAAAyyssrdb
eijkkvvvxxxzzzzyyssrdb
eijkkvvvwwwzzzzyyssrdb
eijkkvvvwwwzzzzyyssrdb
eijkkvvvwwwzzzzyyssrdb
eijkkvvvwwwttttttttrdb
eijkkvvvwwwttttttttrdb
eijkkhhhhhhhhhhhhhhhhb
eijffffffffffffffffffb
eijgggggggggcccccccccc
eijgggggggggcccccccccc

a(23)=6; 16 rectangles;
aaaaaaaaaaaaaaaaaeeeebb
aaaaaaaaaaaaaaaaaeeeebb
ppookkkkkkkiiiiiieeeebb
ppookkkkkkkiiiiiieeeebb
ppookkkkkkkiiiiiieeeebb
ppookkkkkkkiiiiiieeeebb
ppookkkkkkkiiiiiieeeebb
ppoollljjjjjjjjffeeeebb
ppoollljjjjjjjjffggggbb
ppoollljjjjjjjjffggggbb
ppoollljjjjjjjjffggggbb
ppoolllmmmmmhhhffggggbb
ppoolllmmmmmhhhffggggbb
ppoolllmmmmmhhhffggggbb
ppoolllmmmmmhhhffggggbb
ppoolllmmmmmhhhffggggbb
ppoolllmmmmmhhhffggggbb
ppnnnnnnnnnnhhhffcccccc
ppnnnnnnnnnnhhhffcccccc
ppnnnnnnnnnnhhhffcccccc
ddddddddddddhhhffcccccc
ddddddddddddhhhffcccccc
ddddddddddddhhhffcccccc

a(23)=6; 25 rectangles;
aaaaaaaaaaaaaaaaaaaaaab
eeeeeeeeeeeeeeeeeeggfdb
nnoooommmmmmmmmmmmggfdb
nnoooommmmmmmmmmmmggfdb
nnooootttttxxyyyvvggfdb
nnooootttttxxyyyvvggfdb
nnooootttttxxyyyvvggfdb
nnooootttttxxyyyvvggfdb
nnssssssuuuxxyyyvvggfdb
nnssssssuuuxxyyyvvggfdb
nnssssssuuuxxyyyvvggfdb
nnssssssuuuxxyyyvvggfdb
qqqqqqqquuuxxwwwvvggfdb
qqqqqqqquuuxxwwwllllfdb
qqqqqqqquuuxxwwwllllfdb
pppppprrrrrrrwwwllllfdb
pppppprrrrrrrwwwllllfdb
pppppprrrrrrrwwwllllfdb
jjjjjjjjjjjjjjjjjjjjfdb
kkkkkkkkkkkiiiiiiiiiidb
kkkkkkkkkkkiiiiiiiiiidb
hhhhhhhhhhhhhhhhhhhhhdb
ccccccccccccccccccccccc

a(24)=6; 10 rectangles;
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggjjjjjeeeeeeeeeeeebbbb
gggjjjjjeeeeeeeeeeeebbbb
gggjjjjjeeeeeeeeeeeebbbb
gggjjjjjeeeeeeeeeeeebbbb
gggjjjjjeeeeeeeeeeeebbbb
gggjjjjjiiiiiiiiiicccccc
gggjjjjjiiiiiiiiiicccccc
gggjjjjjiiiiiiiiiicccccc
gggjjjjjiiiiiiiiiicccccc
gggjjjjjiiiiiiiiiicccccc
gggjjjjjiiiiiiiiiicccccc
ggghhhhhhhhhhhhhhhcccccc
ggghhhhhhhhhhhhhhhcccccc
ggghhhhhhhhhhhhhhhcccccc
ggghhhhhhhhhhhhhhhcccccc

a(24)=6; 27 rectangles;
aaaaaaaaaaaaaaaaaaaaaaab
fjjjjjjjjjjjjjjjjjjjjjdb
fkkkkkkkkkkkkkkkkkkkkmdb
fhlllllllllllllllllllmdb
fhipppssssuuuuuuuuuvvmdb
fhipppssssuuuuuuuuuvvmdb
fhipppsssstttttttwwvvmdb
fhipppsssstttttttwwvvmdb
fhipppsssstttttttwwvvmdb
fhippprrrrrrrrAAAwwvvmdb
fhippprrrrrrrrAAAwwvvmdb
fhippprrrrrrrrAAAwwvvmdb
fhiggqqqqxxxxxAAAwwvvmdb
fhiggqqqqxxxxxAAAwwvvmdb
fhiggqqqqxxxxxAAAwwvvmdb
fhiggqqqqxxxxxAAAwwvvmdb
fhiggqqqqzzzzzzzzzzzzmdb
fhiggqqqqzzzzzzzzzzzzmdb
fhiggooooooyyyyyyyyyymdb
fhiggooooooyyyyyyyyyymdb
fhiggoooooonnnnnnnnnnndb
fhiggoooooonnnnnnnnnnndb
fhiggeeeeeeeeeeeeeeeeeeb
cccccccccccccccccccccccc

a(25)=6; 14 rectangles;
aaaaaaaaaaaaaaeeeeeefffbb
aaaaaaaaaaaaaaeeeeeefffbb
aaaaaaaaaaaaaaeeeeeefffbb
ddhhhgggggggggeeeeeefffbb
ddhhhgggggggggeeeeeefffbb
ddhhhgggggggggeeeeeefffbb
ddhhhgggggggggeeeeeefffbb
ddhhhgggggggggeeeeeefffbb
ddhhhmmmmmmmmlllllllfffbb
ddhhhmmmmmmmmlllllllfffbb
ddhhhmmmmmmmmlllllllfffbb
ddhhhmmmmmmmmlllllllfffbb
ddhhhmmmmmmmmlllllllfffbb
ddhhhmmmmmmmmlllllllfffbb
ddhhhnnnnnnnnnnnnkkkkkkbb
ddhhhnnnnnnnnnnnnkkkkkkbb
ddhhhnnnnnnnnnnnnkkkkkkbb
ddhhhnnnnnnnnnnnnkkkkkkbb
ddjjjjjjjjjjjjjjjkkkkkkbb
ddjjjjjjjjjjjjjjjkkkkkkbb
ddjjjjjjjjjjjjjjjkkkkkkbb
ddiiiiiiiiiiiiiiiiiiiiibb
ddiiiiiiiiiiiiiiiiiiiiibb
ddccccccccccccccccccccccc
ddccccccccccccccccccccccc

a(26)=6; 15 rectangles;
aaaaaaaaaaaaaaaaaaaaaabbbb
aaaaaaaaaaaaaaaaaaaaaabbbb
ddgggggggggggggghhhhhhbbbb
ddgggggggggggggghhhhhhbbbb
ddgggggggggggggghhhhhhbbbb
ddfflllkkkkkkkkkhhhhhhbbbb
ddfflllkkkkkkkkkhhhhhhbbbb
ddfflllkkkkkkkkkhhhhhhbbbb
ddfflllkkkkkkkkkhhhhhhbbbb
ddfflllkkkkkkkkkhhhhhhbbbb
ddffllliiiiiiiiiiiiiiibbbb
ddffllliiiiiiiiiiiiiiijjjj
ddffllliiiiiiiiiiiiiiijjjj
ddffllloooooooonnnnnnnjjjj
ddffllloooooooonnnnnnnjjjj
ddffllloooooooonnnnnnnjjjj
ddffllloooooooonnnnnnnjjjj
ddffllloooooooonnnnnnnjjjj
ddffllloooooooonnnnnnnjjjj
ddffeeeeeemmmmmmmmmmmmjjjj
ddffeeeeeemmmmmmmmmmmmjjjj
ddffeeeeeemmmmmmmmmmmmjjjj
ddffeeeeeemmmmmmmmmmmmjjjj
ddffeeeeeecccccccccccccccc
ddffeeeeeecccccccccccccccc
ddffeeeeeecccccccccccccccc

a(26)=6; 32 rectangles;
aaaaaaaaaaaaaaaaaaaaaaaggb
eeeeeeeeeeeeeeeeeeetttoggb
djjjjjjjjjjjjuuuuuutttoggb
djjjjjjjjjjjjuuuuuutttoggb
dfknnnnnnnnnnuuuuuutttoggb
dfknnnnnnnnnnvvvvwwtttoggb
dfklmyyyzzzzzvvvvwwtttoggb
dfklmyyyzzzzzvvvvwwtttoggb
dfklmyyyzzzzzvvvvwwxxxoggb
dfklmyyyzzzzzvvvvwwxxxoppb
dfklmyyyFFFFFFFFFwwxxxoppb
dfklmyyyFFFFFFFFFwwxxxoppb
dfklmssDDDDEEEEEEwwxxxoppb
dfklmssDDDDEEEEEEwwxxxoppb
dfklmssDDDDEEEEEEwwxxxoppb
dfklmssDDDDEEEEEEwwxxxoppb
dfklmssDDDDBBBBBBBBBBBoppb
dfklmssDDDDBBBBBBBBBBBoppb
dfklmssCCCCCCCCAAAAAAAoppb
dfklmssCCCCCCCCAAAAAAAoppb
dfklmssCCCCCCCCAAAAAAAoppb
dfklmssrrrrrrrrrrrrrrrrrrb
dfklmqqqqqqqqqqqqqqqqqqqqb
dfklmhhhhhhhhhhhhhhhhhhhhh
dfkliiiiiiiiiiiiiiiiiiiiii
dfcccccccccccccccccccccccc

a(27)=7; 19 rectangles;
aaaaaaaaaaaaaaaaaaabbbbbbbb
aaaaaaaaaaaaaaaaaaabbbbbbbb
gghhhhheeeeeeeeeeeebbbbbbbb
gghhhhheeeeeeeeeeeebbbbbbbb
gghhhhheeeeeeeeeeeebbbbbbbb
gghhhhhdddddddddddddddddddd
gghhhhhdddddddddddddddddddd
gghhhhhiiiiiiiiiiiiiiiiiicc
gghhhhhiiiiiiiiiiiiiiiiiicc
ggooooooooooooooooooooojjcc
ggooooooooooooooooooooojjcc
ggnnnpppppppppssssssqqqjjcc
ggnnnpppppppppssssssqqqjjcc
ggnnnpppppppppssssssqqqjjcc
ggnnnpppppppppssssssqqqjjcc
ggnnnmmmmmllllssssssqqqjjcc
ggnnnmmmmmllllssssssqqqjjcc
ggnnnmmmmmllllssssssqqqjjcc
ggnnnmmmmmllllrrrrrrqqqjjcc
ggnnnmmmmmllllrrrrrrqqqjjcc
ggnnnmmmmmllllrrrrrrqqqjjcc
ggnnnmmmmmllllrrrrrrqqqjjcc
ggnnnmmmmmllllrrrrrrqqqjjcc
ffffffffffllllrrrrrrqqqjjcc
ffffffffffkkkkkkkkkkkkkjjcc
ffffffffffkkkkkkkkkkkkkjjcc
ffffffffffkkkkkkkkkkkkkjjcc

a(27)=7; 30 rectangles;
aaaaaaaaaaaaaaaaaaaaaaaaaab
egjifdddddddddddddddddddddb
egjifooooooooppnnnnnnnnnnnb
egjifooooooooppnnnnnnnnnnnb
egjifooooooooppAAzzzzzzzzzb
egjifrrrrrrqqppAAzzzzzzzzzb
egjifrrrrrrqqppAAzzzzzzzzzb
egjiftttsssqqppAACCCCBBByyb
egjiftttsssqqppAACCCCBBByyb
egjiftttsssqqppAACCCCBBByyb
egjiftttsssqqppAACCCCBBByyb
egjiftttsssqqppAACCCCBBByyb
egjiftttsssqqppAACCCCBBByyb
egjiftttvvvvvvvvvvvvvBBByyb
egjiftttvvvvvvvvvvvvvBBByyb
egjifuuuuuuuxxxxxxxxxxxxyyb
egjifuuuuuuuxxxxxxxxxxxxyyb
egjifuuuuuuuwwwwwwwwwwwwwwb
egjifuuuuuuuwwwwwwwwwwwwwwb
egjimmmmmmmmmmmmmmmmmmmmmmb
egjlllllllllllllllllllllllb
egkkkkkkkkkkkkkkkkkkkkkkkkb
ehhhhhhhhhhhhhhhhhhhhhhhhhb
ccccccccccccccccccccccccccc

a(27)=7; 31 rectangles;
aaaaaaaaaaaaaaaaaaaaaaaaaab
egfdddddddddddddddddddddddb
egfklljjjjjjjjjjjjjjjjjjjjb
egfkllqqyyyyyyyyyyyCCCCAApb
egfkllqqyyyyyyyyyyyCCCCAApb
egfkllqqxxxDDDDDDDDCCCCAApb
egfkllqqxxxDDDDDDDDCCCCAApb
egfkllqqxxxDDDDDDDDCCCCAApb
egfkllqqxxxEEEEEEEEEEEEAApb
egfkllqqxxxEEEEEEEEEEEEAApb
egfkllqqxxxzzzzzwwwBBBBAApb
egfkllqqxxxzzzzzwwwBBBBAApb
egfkllqqxxxzzzzzwwwBBBBAApb
egfkllqqxxxzzzzzwwwBBBBAApb
egfknnnssssssssswwwBBBBAApb
egfknnnssssssssswwwBBBBAApb
egfknnnssssssssswwwvvvvvvpb
egfknnnrrrrruuuuuuuvvvvvvpb
egfknnnrrrrruuuuuuuvvvvvvpb
egfknnnrrrrruuuuuuuvvvvvvpb
egfknnnrrrrrtttttttttttttpb
egfknnnrrrrrtttttttttttttpb
egfkooooooooooooooooooooopb
egfkmmmmmmmmmmmmmmmmmmmmmmb
egiiiiiiiiiiiiiiiiiiiiiiiib
ehhhhhhhhhhhhhhhhhhhhhhhhhb
ccccccccccccccccccccccccccc

a(27)=7; 34 rectangles;
aaaaaaaaaaaaaaaaaaaaaaaaghb
eeeeeeeeeeeeeeeeeeeqqqrpghb
duuuuuuuuuuuuuuuuuuqqqrpghb
dwwwwwwwwwwwwxxxxxxqqqrpghb
dwwwwwwwwwwwwxxxxxxqqqrpghb
dvvvvvvvvvyyyxxxxxxqqqrpghb
dvvvvvvvvvyyyxxxxxxqqqrpghb
dfoonzzAAAyyyDDDDEEEEErpghb
dfoonzzAAAyyyDDDDEEEEErpghb
dfoonzzAAAyyyDDDDEEEEErpghb
dfoonzzAAAyyyDDDDEEEEErpghb
dfoonzzAAAyyyDDDDEEEEErpghb
dfoonzzAAACCCCCCCCFFFFrpghb
dfoonzzAAACCCCCCCCFFFFrpghb
dfoonzzAAACCCCCCCCFFFFrpghb
dfoonzzBBBBBHHHHHHFFFFrpghb
dfoonzzBBBBBHHHHHHFFFFrpghb
dfmmnzzBBBBBHHHHHHFFFFrpghb
dfmmnzzBBBBBGGGGGGGGGGrpghb
dfmmntttttttGGGGGGGGGGrpghb
dfmmntttttttssssssssssspghb
dfmmntttttttssssssssssspghb
dfmmnllllllllllllllllllllhb
dfmmnjjjjjjjjjjjjjjjjjjjjjb
dfmmniiiiiiiiiiiiiiiiiiiiii
dfmmkkkkkkkkkkkkkkkkkkkkkkk
dfccccccccccccccccccccccccc

a(28)=7; 32 rectangles;
aaaaaaaaaaaaaaaaaaaaaaaaaaab
enppmmdddddddddddddddddddddb
enppmmqqqrrrrlllllllllllljgb
enppmmqqqrrrrlllllllllllljgb
enppmmqqqrrrrtttttttuusskjgb
enppmmqqqrrrrtttttttuusskjgb
enppmmqqqrrrrtttttttuusskjgb
enppmmqqqrrrrwwwwvvvuusskjgb
enppmmqqqrrrrwwwwvvvuusskjgb
enppmmxxxxxxxwwwwvvvuusskjgb
enppmmxxxxxxxwwwwvvvuusskjgb
enppmmxxxxxxxwwwwvvvuusskjgb
enppmmxxxxxxxwwwwvvvuusskjgb
enppyyyyyyyyyyyyyvvvuusskjgb
enppyyyyyyyyyyyyyvvvuusskjgb
enzzzzzzEEEEEEEEEEEEEEsskjgb
enzzzzzzEEEEEEEEEEEEEEsskjgb
enzzzzzzCCCCCCCCCCCDDDDDkjgb
enzzzzzzCCCCCCCCCCCDDDDDkjgb
enBBBBBBBBFFFFFFFFFDDDDDkjgb
enBBBBBBBBFFFFFFFFFDDDDDkjgb
enBBBBBBBBFFFFFFFFFDDDDDkjgb
enAAAAAAAAAAAAAAAAAAAAAAkjgb
eoooooooooooooooooooooookjgb
hhhhhhhhhhhhhhhhhhhhhhhhkjgb
iiiiiiiiiiiiiiiiiiiiiiiiijgb
ffffffffffffffffffffffffffgb
cccccccccccccccccccccccccccc

a(29)=7; 34 rectangles;
aaaaaaaaaaaaaaaaaaaaaaaaaaefb
dilllllllllllllllllllllllmefb
diknnnnnnnnnnnnnnnnnnnnnnmefb
dikjtttttwwwwwwwwwwwwwwxxmefb
dikjtttttwwwwwwwwwwwwwwxxmefb
dikjtttttvvvvvvvAABBBzzxxmefb
dikjtttttvvvvvvvAABBBzzxxmefb
dikjtttttvvvvvvvAABBBzzxxmefb
dikjuuuuuuuuDDDDAABBBzzxxmefb
dikjuuuuuuuuDDDDAABBBzzxxmefb
dikjuuuuuuuuDDDDAABBBzzxxmefb
dikjssEEEEEEDDDDAABBBzzxxmefb
dikjssEEEEEEDDDDAABBBzzxxmefb
dikjssEEEEEEDDDDAACCCzzxxmefb
dikjssEEEEEEDDDDAACCCzzxxmefb
dikjssGGGHHHHHHHAACCCzzxxmefb
dikjssGGGHHHHHHHAACCCyyyymefb
dikjssGGGHHHHHHHAACCCyyyymefb
dikjssGGGHHHHHHHAACCCyyyymefb
dikjssGGGFFFFFFFFFCCCyyyymefb
dikjssGGGFFFFFFFFFCCCyyyymefb
dikjssGGGFFFFFFFFFCCCyyyymefb
dikjssqqqqqqqqqqqqqqqqqqqqqfb
dikjrrrrrrrrrrrrrpppppppppppb
dikjrrrrrrrrrrrrrpppppppppppb
dikjoooooooooooooooooooooooob
dikjggggggggggggggggggggggggg
dihhhhhhhhhhhhhhhhhhhhhhhhhhh
dcccccccccccccccccccccccccccc

 2017-01-05, 17:56 #41 EdPeggJr   Nov 2016 24 Posts Many of those are unexpectedly wild. I didn't foresee the variant solution for 18 to have such a range. I imagine the ranges for the standard problem will be much tamer by comparison.
 2017-01-07, 00:22 #42 EdPeggJr   Nov 2016 24 Posts Your solution for a(26) here uses each rectangle exactly twice, with all in different orientations. I have a similar solution for a(10), and no others.
 2017-09-17, 15:44 #43 Elgers Niels   Sep 2017 Belgium 12 Posts Please consider posting your results on this website, where you can compete for the best tiling.
 2018-06-17, 23:32 #44 shanem   Jun 2018 1 Posts It may be a good idea to keep track of the amount of rectangles used to create the squares. The values for the area will most likely be close to the value of the area of the square divided by the number of squares as we are looking for the closest values of area. This could potentially lead to some sort of correlation.

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