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Old 2016-12-02, 20:34   #23
EdPeggJr
 
Nov 2016

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One amusing method for tackling large squares occurred to me.

1. Compile a catalog of perfect k-rectangle Blanche dissections of the unit square for various k from 7 to 16. Each rectangle in a dissection will have area of exactly 1/k.

2. Grade the various Blanche dissections by the relative closeness of component rectangles.

3. Recursion. Take an order j Blanche dissection and map various high grade order k Blanche dissections to each rectangle. That will divide the unit square into j*k non-congruent rectangles each with area of exactly 1/(j*k).

4. Multiply and round. Find integer multiples that yield non-congruent rectangles.
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Old 2016-12-05, 08:59   #24
R. Gerbicz
 
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Quote:
Originally Posted by EdPeggJr View Post
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}}
(...)
For the first case: it is impossible to pack that.
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Old 2016-12-08, 04:25   #25
EdPeggJr
 
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Sequence https://oeis.org/A278970 is known up to a(44). Here are currently known best values up to 120.
x, x, 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, 3| 1, 1, 1, 0, 0, 1,
1, 1, 1, 1, 1, 2, 1, 2, 3, 0,
a, 3, 1, 4, 1, 0, 1, 1, 2, 0,
2, 0, 1, 1, 1, 2, 1, a, 1, 2,
1, 2, 2, 0, 1, 5, 1, 3, 1, 0,
1, 0, 0, 0, 0, 1, 1, 2, b, 0,
1, 1, 2, 1, 2, a, a, 3, 0, 0,
0, 0, 0, 1, a, b, 1, 0, 0, a.
An "a" represents -1. Currently at 61, 78, 99, 106, 107, 115, 120.
A "b" represents -2. Currently at 99 and 116.
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Old 2016-12-08, 16:49   #26
EdPeggJr
 
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What a difference a good overnight run can make. I've found a good solution for square 61, and many other squares. Current values to a(120) are as follows:

x, x, 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, 3, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 0,
3, 3, 1, 4, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, -1, 1, 2,
1, 2, 2, 1, 1, 5, 1, 3, 1, 0, 1, 2, 2, 0, 0, 1, 1, 2, -2, 0,
1, 1, 2, 1, 2, -1, -1, 3, 0, 0, 0, 0, 0, 1, -1, -2, 1, 0, 0, -1

I've updated http://math.stackexchange.com/questi...und-for-defect and included a plot. The connection to n/log(n) for this problem still seems wild to me. Squares 78, 99, 106, 107, 115, 116, 120 are still negative.

My square 61 has defect 15, areas 238 to 253.
My square 74 has defect 20, areas 414 to 434.
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Old 2016-12-12, 16:49   #27
EdPeggJr
 
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In the variant problem, divide a square of size N into rectangles so that no rectangles are translations of each other. Minimize the difference between the largest area and smallest area. Basically, it's the same as the Mondrian problem, but a rectangle can be reused if it has a different orientation. To 120, here are my best results so far for minimal possible defects.

0, 0, 2, 2, 4, 3, 3, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 6,
6, 6, 7, 6, 7, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8, 8, 11, 10, 10, 10,
11, 11, 11, 12, 10, 10, 12, 11, 12, 12, 8, 12, 12, 14, 12, 14, 14, 14, 15, 12,
15, 14, 15, 14, 16, 16, 15, 16, 16, 16, 17, 16, 18, 14, 20, 18, 16, 18, 16, 18,
19, 16, 20, 18, 16, 18, 17, 19, 22, 17, 17, 21, 20, 20, 22, 23, 24, 22, 23, 24,
21, 20, 14, 23, 24, 22, 27, 24, 25, 25, 24, 27, 24, 27, 29, 27, 24, 28, 25, 24

Can anyone verify / improve these? http://demonstrations.wolfram.com/MondrianArtProblem/ has a compilation of results.
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Old 2016-12-13, 22:42   #28
R. Gerbicz
 
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Quote:
Originally Posted by EdPeggJr View Post
In the variant problem, divide a square of size N into rectangles so that no rectangles are translations of each other.
[...]
Can anyone verify / improve these? http://demonstrations.wolfram.com/MondrianArtProblem/ has a compilation of results.
I've found the optimal values for n=3..27
Code:
// rectangle's different orientation is permitted
a(3)=2
aab
ccb
ccb

a(4)=2
aaab
deeb
deeb
cccc

a(5)=3
aaaab
dddeb
fggeb
fggeb
ccccc

a(6)=3
aaaaab
dddefb
hhhefb
hhhefb
ggggfb
cccccc

a(7)=3
aaaaabb
djihhbb
djihhbb
djifffe
djifffe
djgggge
dcccccc

a(8)=3
aaaaaabb
ghhiiibb
ghhiiibb
ghhiiibb
gddddddd
gffffeee
gffffeee
cccccccc

a(9)=3
aaaaaaaab
ddddddeeb
fggghheeb
fggghheeb
fiiihheeb
fiiijjjjb
fiiijjjjb
fkkkkkkkb
ccccccccc

a(10)=4
aaaaaaaaab
ddddddeefb
jiiihheefb
jiiihheefb
jiiihheefb
jgggggggfb
jlllmmmmfb
jlllmmmmfb
kkkkkkkkfb
cccccccccc

a(11)=4
aaaaaaaaaab
ddddddddeeb
hjjjkkkkeeb
hjjjkkkkeeb
hjjjlllleeb
hjjjlllleeb
hiiilllleeb
hiiiffffffb
hiiiffffffb
hgggggggggb
ccccccccccc

a(12)=4
aaaaaaaaaaab
ddddddddeeeb
fgggghhheeeb
fgggghhheeeb
fiijjhhheeeb
fiijjnnnnmmb
fiijjnnnnmmb
fiijjnnnnmmb
fiillllllmmb
fiillllllmmb
fkkkkkkkkkkb
cccccccccccc

a(13)=4
aaaaaaaaaaabb
dhjjjjjjjjjbb
dhimlllllkkbb
dhimlllllkkbb
dhimpppookkbb
dhimpppookkbb
dhimpppooeeee
dhimpppooeeee
dhimnnnooeeee
dhimnnnffffff
dhimnnnffffff
dhigggggggggg
dcccccccccccc

a(14)=4
aaaaaaaeeeffbb
aaaaaaaeeeffbb
aaaaaaaeeeffbb
ddiiiiieeeffbb
ddiiiiieeeffbb
ddiiiiieeeffbb
ddiiiiieeeffbb
ddhhhhhhhhffbb
ddhhhhhhhhffbb
ddhhhhhhhhffbb
ddggggggggggbb
ddggggggggggbb
ddcccccccccccc
ddcccccccccccc

a(15)=5
aaaaaaaaaaaaaab
ddddddddddeeeeb
fgmmlllllleeeeb
fgmmlllllleeeeb
fgmmrrrkkkkkkkb
fgmmrrrkkkkkkkb
fgmmrrrppnnnnnb
fgmmrrrppnnnnnb
fgmmrrrppqqqjjb
fgoooooppqqqjjb
fgoooooppqqqjjb
fgoooooppqqqjjb
fhhhhhhhhhhhjjb
fiiiiiiiiiiiiib
ccccccccccccccc

a(16)=5
aaaaaaaaaaaaaaab
dddddddddddeeefb
ghjlllllllleeefb
ghjlllllllleeefb
ghjmmmmmmnneeefb
ghjmmmmmmnnooofb
ghjssrrrrnnooofb
ghjssrrrrnnooofb
ghjssrrrrnnooofb
ghjssrrrrnnooofb
ghjssqqqqqppppfb
ghjssqqqqqppppfb
ghjssqqqqqppppfb
ghkkkkkkkkkkkkfb
giiiiiiiiiiiiifb
cccccccccccccccc

a(17)=5
aaaaaaaaaaaaabbbb
aaaaaaaaaaaaabbbb
aaaaaaaaaaaaabbbb
eeeffffggggggbbbb
eeeffffggggggbbbb
eeeffffggggggbbbb
eeeffffggggggbbbb
eeeffffggggggbbbb
eeeffffggggggbbbb
eeeffffggggggbbbb
eeeffffdddddddddd
eeeffffdddddddddd
eeeffffdddddddddd
eeeffffdddddddddd
eeecccccccccccccc
eeecccccccccccccc
eeecccccccccccccc

a(18)=6
aaaaaaeeeeeeeeeebb
aaaaaaeeeeeeeeeebb
aaaaaaeeeeeeeeeebb
aaaaaagggiijjfffbb
ddhhhhgggiijjfffbb
ddhhhhgggiijjfffbb
ddhhhhgggiijjfffbb
ddhhhhgggiijjfffbb
ddhhhhgggiijjfffbb
ddhhhhgggiijjfffbb
ddhhhhgggiijjfffbb
ddllllllliijjfffbb
ddllllllliijjfffbb
ddllllllliijjccccc
ddllllllliijjccccc
ddkkkkkkkkkjjccccc
ddkkkkkkkkkjjccccc
ddkkkkkkkkkjjccccc

a(19)=6
aaaaaaaaaaeeeeeeebb
aaaaaaaaaaeeeeeeebb
aaaaaaaaaaeeeeeeebb
ddffffffffeeeeeeebb
ddffffffffeeeeeeebb
ddffffffffgggghhhbb
ddffffffffgggghhhbb
ddjjjkkkkkgggghhhbb
ddjjjkkkkkgggghhhbb
ddjjjkkkkkgggghhhbb
ddjjjkkkkkgggghhhbb
ddjjjkkkkkgggghhhbb
ddjjjkkkkkgggghhhbb
ddjjjiiiiiiiiihhhbb
ddjjjiiiiiiiiihhhbb
ddjjjiiiiiiiiihhhbb
ddjjjiiiiiiiiihhhbb
ddccccccccccccccccc
ddccccccccccccccccc

a(20)=4
aaaaaaaaaaaaaaaabbbb
aaaaaaaaaaaaaaaabbbb
aaaaaaaaaaaaaaaabbbb
eeegggggggggggggbbbb
eeegggggggggggggbbbb
eeegggggggggggggbbbb
eeegggggggggggggbbbb
eeefffffhhhhhhhhbbbb
eeefffffhhhhhhhhbbbb
eeefffffhhhhhhhhbbbb
eeefffffhhhhhhhhbbbb
eeefffffhhhhhhhhbbbb
eeefffffhhhhhhhhbbbb
eeefffffcccccccccccc
eeefffffcccccccccccc
eeefffffcccccccccccc
eeefffffcccccccccccc
eeeddddddddddddddddd
eeeddddddddddddddddd
eeeddddddddddddddddd

a(21)=6
aaaaaaaaaaaaaaaabbbbb
aaaaaaaaaaaaaaaabbbbb
eeeffffffgggggggbbbbb
eeeffffffgggggggbbbbb
eeeffffffgggggggbbbbb
eeeffffffgggggggbbbbb
eeeffffffgggggggbbbbb
eeeiiiiiiiiiiiiiiiiii
eeeiiiiiiiiiiiiiiiiii
eeejjjjjjjjjjjjkkkkkk
eeejjjjjjjjjjjjkkkkkk
eeejjjjjjjjjjjjkkkkkk
ddddlllllllllllkkkkkk
ddddlllllllllllkkkkkk
ddddlllllllllllkkkkkk
ddddmmmmmmmmmmmmmmmmm
ddddmmmmmmmmmmmmmmmmm
ddddhhhhhhhhccccccccc
ddddhhhhhhhhccccccccc
ddddhhhhhhhhccccccccc
ddddhhhhhhhhccccccccc

a(22)=6
aaaaaaaaaaaaaaaaabbbbb
aaaaaaaaaaaaaaaaabbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggffffffffeeeeeeebbbbb
ggiiijjjjddddddddddddd
ggiiijjjjddddddddddddd
ggiiijjjjddddddddddddd
ggiiijjjjmmmmmmllllccc
ggiiijjjjmmmmmmllllccc
ggiiijjjjmmmmmmllllccc
ggiiijjjjmmmmmmllllccc
ggiiijjjjmmmmmmllllccc
ggiiijjjjmmmmmmllllccc
ggiiikkkkkkkkkkllllccc
ggiiikkkkkkkkkkllllccc
ggiiikkkkkkkkkkllllccc
ggiiikkkkkkkkkkllllccc
hhhhhhhhhhhhhhhhhhhccc
hhhhhhhhhhhhhhhhhhhccc

a(23)=6
aaaaaaaaaaaaaaaaabbbbbb
aaaaaaaaaaaaaaaaabbbbbb
llmmmjjjjjjjjjjjjbbbbbb
llmmmjjjjjjjjjjjjbbbbbb
llmmmjjjjjjjjjjjjbbbbbb
llmmmkkkkkkkkkkiibbbbbb
llmmmkkkkkkkkkkiiffeeee
llmmmkkkkkkkkkkiiffeeee
llmmmoooooppphhiiffeeee
llmmmoooooppphhiiffeeee
llmmmoooooppphhiiffeeee
llmmmoooooppphhiiffeeee
llmmmoooooppphhiiffeeee
llmmmoooooppphhiiffeeee
llnnnnnnnnppphhiiffcccc
llnnnnnnnnppphhiiffcccc
llnnnnnnnnppphhiiffcccc
llnnnnnnnnppphhiiffcccc
dddddddgggggghhiiffcccc
dddddddgggggghhiiffcccc
dddddddgggggghhiiffcccc
dddddddgggggghhiiffcccc
dddddddgggggghhiiffcccc

a(24)=6
aaaaaaaaaaaaaaddddddbbbb
aaaaaaaaaaaaaaddddddbbbb
aaaaaaaaaaaaaaddddddbbbb
aaaaaaaaaaaaaaddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
gggfffffffffffddddddbbbb
ggghhhhheeeeeeeeeeeebbbb
ggghhhhheeeeeeeeeeeebbbb
ggghhhhheeeeeeeeeeeebbbb
ggghhhhheeeeeeeeeeeebbbb
ggghhhhheeeeeeeeeeeebbbb
ggghhhhhjjjjjjjjjjcccccc
ggghhhhhjjjjjjjjjjcccccc
ggghhhhhjjjjjjjjjjcccccc
ggghhhhhjjjjjjjjjjcccccc
ggghhhhhjjjjjjjjjjcccccc
ggghhhhhjjjjjjjjjjcccccc
gggiiiiiiiiiiiiiiicccccc
gggiiiiiiiiiiiiiiicccccc
gggiiiiiiiiiiiiiiicccccc
gggiiiiiiiiiiiiiiicccccc

a(25)=6
aaaaaaaaaaaaaaeeeeeefffbb
aaaaaaaaaaaaaaeeeeeefffbb
aaaaaaaaaaaaaaeeeeeefffbb
ddkkkjjjjjjjjjeeeeeefffbb
ddkkkjjjjjjjjjeeeeeefffbb
ddkkkjjjjjjjjjeeeeeefffbb
ddkkkjjjjjjjjjeeeeeefffbb
ddkkkjjjjjjjjjeeeeeefffbb
ddkkkmmmmmmmmiiiiiiifffbb
ddkkkmmmmmmmmiiiiiiifffbb
ddkkkmmmmmmmmiiiiiiifffbb
ddkkkmmmmmmmmiiiiiiifffbb
ddkkkmmmmmmmmiiiiiiifffbb
ddkkkmmmmmmmmiiiiiiifffbb
ddkkknnnnnnnnnnnnggggggbb
ddkkknnnnnnnnnnnnggggggbb
ddkkknnnnnnnnnnnnggggggbb
ddkkknnnnnnnnnnnnggggggbb
ddlllllllllllllllggggggbb
ddlllllllllllllllggggggbb
ddlllllllllllllllggggggbb
ddhhhhhhhhhhhhhhhhhhhhhbb
ddhhhhhhhhhhhhhhhhhhhhhbb
ddccccccccccccccccccccccc
ddccccccccccccccccccccccc

a(26)=6
aaaaaaaaaaaaaaaaaaaaaabbbb
aaaaaaaaaaaaaaaaaaaaaabbbb
ddffffffffffffffggggggbbbb
ddffffffffffffffggggggbbbb
ddffffffffffffffggggggbbbb
ddeeiiiiiiiiiiiiggggggbbbb
ddeeiiiiiiiiiiiiggggggbbbb
ddeeiiiiiiiiiiiiggggggbbbb
ddeeiiiiiiiiiiiiggggggbbbb
ddeejjjjjjjjjjjjjjjkkkbbbb
ddeejjjjjjjjjjjjjjjkkkbbbb
ddeejjjjjjjjjjjjjjjkkkbbbb
ddeemmmmmmmooooooookkkllll
ddeemmmmmmmooooooookkkllll
ddeemmmmmmmooooooookkkllll
ddeemmmmmmmooooooookkkllll
ddeemmmmmmmooooooookkkllll
ddeemmmmmmmooooooookkkllll
ddeehhhhhhnnnnnnnnnkkkllll
ddeehhhhhhnnnnnnnnnkkkllll
ddeehhhhhhnnnnnnnnnkkkllll
ddeehhhhhhnnnnnnnnnkkkllll
ddeehhhhhhnnnnnnnnnkkkllll
ddeehhhhhhcccccccccccccccc
ddeehhhhhhcccccccccccccccc
ddeehhhhhhcccccccccccccccc

a(27)=7
aaaaaaaaaaaaaaaaaaabbbbbbbb
aaaaaaaaaaaaaaaaaaabbbbbbbb
gghhhhhffffffffffffbbbbbbbb
gghhhhhffffffffffffbbbbbbbb
gghhhhhffffffffffffbbbbbbbb
gghhhhheeeeeeeeeeeeeeeeeeee
gghhhhheeeeeeeeeeeeeeeeeeee
gghhhhhiiiiiiiiiiiiiiiiiidd
gghhhhhiiiiiiiiiiiiiiiiiidd
ggjjjjjjjjjjjjjjjjjjjjjkkdd
ggjjjjjjjjjjjjjjjjjjjjjkkdd
gglllmmmmmmmmmqqqqqqrrrkkdd
gglllmmmmmmmmmqqqqqqrrrkkdd
gglllmmmmmmmmmqqqqqqrrrkkdd
gglllmmmmmmmmmqqqqqqrrrkkdd
gglllnnnnnooooqqqqqqrrrkkdd
gglllnnnnnooooqqqqqqrrrkkdd
gglllnnnnnoooossssssrrrkkdd
gglllnnnnnoooossssssrrrkkdd
gglllnnnnnoooossssssrrrkkdd
gglllnnnnnoooossssssrrrkkdd
gglllnnnnnoooossssssrrrkkdd
gglllnnnnnoooossssssrrrkkdd
ccccccccccoooossssssrrrkkdd
ccccccccccpppppppppppppkkdd
ccccccccccpppppppppppppkkdd
ccccccccccpppppppppppppkkdd
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Old 2016-12-14, 05:32   #29
EdPeggJr
 
Nov 2016

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Great! I did find some of these earlier, and I found 5 and 12 today while I was doing some polishing, but I was far from getting the optimal results for quite a few of them. The current best results up to 80 are

x, x, 2, 2, 3, 3, 3, 3, 3, 4,
4, 4, 4, 4, 5, 5, 5, 6, 6, 4,
6, 6, 6, 6, 6, 6, 7, 8, 8, 8,
9, 8, 8, 8, 8, 8, 11, 10, 10, 10,
11, 11, 11, 12, 10, 10, 12, 11, 12, 12,
8, 12, 12, 13, 12, 14, 14, 14, 15, 12,
15, 14, 15, 14, 16, 16, 15, 16, 16, 16,
17, 16, 18, 14, 18, 18, 16, 18, 16, 18.

It seems to be bounded above by ceiling( n / log(n) ). The distance from that upper bound gives what I call the quality of the dissection. Current values to 80 are.

x, x, 1, 1, 1, 1, 1, 1, 2, 1,
1, 1, 2, 2, 1, 1, 2, 1, 1, 3,
1, 2, 2, 2, 2, 2, 2, 1, 1, 1,
1, 2, 2, 2, 2, 3, 0, 1, 1, 1,
1, 1, 1, 0, 2, 3, 1, 2, 1, 1,
5, 2, 2, 1, 2, 0, 1, 1, 0, 3,
0, 2, 1, 2, 0, 0, 1, 1, 1, 1,
0, 1, 0, 4, 0, 0, 2, 0, 3, 1
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Old 2016-12-16, 04:03   #30
EdPeggJr
 
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https://oeis.org/A279596
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Old 2016-12-28, 01:05   #31
R. Gerbicz
 
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Oct 2005
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My new code really rocks, found the values of a(45)-a(57) for the original puzzle [previously we have known only a(47) in this range]. Made changes at the sequence, and uploaded optimal tilings for n=3..57 at https://oeis.org/A276523/a276523.txt . In two cases used 27 rectangles, so needed to use 'A' also...

Last fiddled with by R. Gerbicz on 2016-12-28 at 01:08 Reason: grammar
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Old 2016-12-28, 21:45   #32
EdPeggJr
 
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The new solutions amaze me. Mostly because I came so close to the correct solutions with my non-optimal code. For 45-57, you gave improvements of 1,2,3,1,4,2,1,3,2,1,1,1,1 over my best solutions. Did having those best-known values help at all?

I need to make code improvements to make more progress on my side. The process of solving a graph bogs down a lot at 18 rectangles, and there are more 18 node graphs than I can process. But it seems like there should be a faster solving method. With the Squared Squares problem, the matrix solutions lead to a shortcut, allowing for much faster solutions. I haven't figured out how to do that. My methods are good for millions, but I need to expand that to billions to make more progress.
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Old 2016-12-29, 19:54   #33
R. Gerbicz
 
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Quote:
Originally Posted by EdPeggJr View Post
The new solutions amaze me. Mostly because I came so close to the correct solutions with my non-optimal code. For 45-57, you gave improvements of 1,2,3,1,4,2,1,3,2,1,1,1,1 over my best solutions. Did having those best-known values help at all?
In my latest codes first I'm searching for deficient=0, if not found a solution then see for deficient=1 etc. until we not find a solution. But your numbers could give us what is an easy problem, there is a large fluctuation in running times: it took 12 minutes (Wall-clock time) to find a(52), but 21 hours to find a(54).

Today found a really good solution for the n=101 big problem from the Numberphile video. Here a very easy and quick search shows that 14<=a(101), and a larger search shows that 19 rectangles doesn't improve our deficient=17.
Code:
a(101)<=17
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
ffffffffffffffffffffffffffffffffffffffffffffffffffffiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjiiiiiiiiiiiiiiiiiiiiiiiiiiiibbbbbbbbbbbbbbbbbbbbb
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
gggggggggggggggggggggggggggggggggggggggggggjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjppppppppppppppooooooooooooooooooooooooooooooooooo
ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn
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ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn
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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn
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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn
ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn
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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn
ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn
ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn
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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn
ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn
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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn
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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn
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