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2016-12-02, 20:34
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#23 |
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Nov 2016
24 Posts |
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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2016-12-05, 08:59
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#24 | |
"Robert Gerbicz"
Oct 2005
Hungary
101110011002 Posts |
Quote:
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2016-12-08, 04:25
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#25 |
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Nov 2016
24 Posts |
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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2016-12-08, 16:49
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#26 |
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Nov 2016
24 Posts |
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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2016-12-12, 16:49
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#27 |
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Nov 2016
1610 Posts |
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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2016-12-13, 22:42
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#28 | |
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"Robert Gerbicz"
Oct 2005
Hungary
22×7×53 Posts |
Quote:
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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2016-12-14, 05:32
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#29 |
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Nov 2016
24 Posts |
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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2016-12-16, 04:03
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#30 |
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Nov 2016
1016 Posts |
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2016-12-28, 01:05
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#31 |
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"Robert Gerbicz"
Oct 2005
Hungary
27148 Posts |
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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2016-12-28, 21:45
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#32 |
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Nov 2016
100002 Posts |
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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2016-12-29, 19:54
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#33 | |
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"Robert Gerbicz"
Oct 2005
Hungary
22×7×53 Posts |
Quote:
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 ddddddddhhhhhhhhhhhhhhhhhhhhhhhhhhkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn 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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjpppppppppppppprrrrrrrrrrrrrrrrrrrrrrrrnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkjjjjjjjjjqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqnnnnnnnnnnn 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ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn ddddddddeeeeeeeeeeeellllllllllllllkkkkkkkkkmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmnnnnnnnnnnn ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc ddddddddeeeeeeeeeeeeccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc |
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