20161127, 20:31  #1 
Aug 2012
Mass., USA
2·3·53 Posts 
Mondrian art puzzles  error in Numberphile video
The recent Numberphile video about Mondrian art puzzles has an error.
It claims that the best Mondrian score for an 18x18 square is 10. But I've achieved a Mondrian score of 8 for it. Code:
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ? ? ? ?
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ? ? ? ?
% % % < < < < < # # # # # # ? ? ? ?
% % % < < < < < # # # # # # ? ? ? ?
% % % < < < < < # # # # # # ? ? ? ?
% % % < < < < < # # # # # # ? ? ? ?
% % % < < < < < # # # # # # ? ? ? ?
% % % < < < < < # # # # # # ? ? ? ?
% % % > > > > > > > + + + + ? ? ? ?
% % % > > > > > > > + + + + = = = =
% % % > > > > > > > + + + + = = = =
% % % > > > > > > > + + + + = = = =
% % % > > > > > > > + + + + = = = =
@ @ @ @ @ @ @ @ @ @ + + + + = = = =
@ @ @ @ @ @ @ @ @ @ + + + + = = = =
@ @ @ @ @ @ @ @ @ @ + + + + = = = =

20161127, 21:11  #3  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
Last fiddled with by science_man_88 on 20161127 at 22:10 

20161128, 19:43  #5 
Aug 2012
Mass., USA
2·3·53 Posts 
Ed has informed me that A276523 has now been corrected. The correction also include new values for 15x15 and 19x19 cases, found by Ed after further checking.

20161128, 20:12  #6 
Aug 2006
1011101100001_{2} Posts 

20161128, 23:33  #7 
"Robert Gerbicz"
Oct 2005
Hungary
2·733 Posts 
Really fascinating puzzle!
My exahaustive code gives the following better solutions (these are optimal): a(14)=6 (!!!) Code:
aaaaaaaaaabbbb aaaaaaaaaabbbb aaaaaaaaaabbbb cccdddddddbbbb cccdddddddbbbb cccdddddddbbbb cccdddddddbbbb cccdddddddbbbb ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff ccceeeeeffffff Code:
aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa bbbbbbbbbbccccdd bbbbbbbbbbccccdd bbbbbbbbbbccccdd eeefffffffccccdd eeefffffffccccdd eeefffffffccccdd eeefffffffccccdd eeefffffffccccdd eeeggggghhhhhhdd eeeggggghhhhhhdd eeeggggghhhhhhdd eeeggggghhhhhhdd eeeggggghhhhhhdd eeeggggghhhhhhdd Code:
aaaaaaaaaaaaaaaaaabbbbb aaaaaaaaaaaaaaaaaabbbbb aaaaaaaaaaaaaaaaaabbbbb aaaaaaaaaaaaaaaaaabbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb cccccccceeeeffffffbbbbb ddddddddeeeeffffffbbbbb ddddddddeeeeffffffbbbbb ddddddddeeeeffffffbbbbb ddddddddeeeeggggggggggg ddddddddeeeeggggggggggg ddddddddeeeeggggggggggg ddddddddeeeeggggggggggg ddddddddeeeeggggggggggg ddddddddeeeeggggggggggg ddddddddeeeeggggggggggg Code:
aaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaa bbbbbbbbbbbbbbbbbbbbccccc bbbbbbbbbbbbbbbbbbbbccccc ddeeeeeeeffffkkkkkkkccccc ddeeeeeeeffffkkkkkkkccccc ddeeeeeeeffffkkkkkkkccccc ddeeeeeeeffffkkkkkkkccccc ddeeeeeeeffffkkkkkkkccccc ddeeeeeeeffffkkkkkkkccccc ddeeeeeeefffflllllllllmmm ddggghhhhfffflllllllllmmm ddggghhhhfffflllllllllmmm ddggghhhhfffflllllllllmmm ddggghhhhfffflllllllllmmm ddggghhhhjjjjjjjjnnnnnmmm ddggghhhhjjjjjjjjnnnnnmmm ddggghhhhjjjjjjjjnnnnnmmm ddggghhhhjjjjjjjjnnnnnmmm ddggghhhhjjjjjjjjnnnnnmmm ddggghhhhjjjjjjjjnnnnnmmm ddgggiiiiiiiiiiiinnnnnmmm ddgggiiiiiiiiiiiinnnnnmmm ddgggiiiiiiiiiiiinnnnnmmm ddgggiiiiiiiiiiiinnnnnmmm Code:
aaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaa bbbbbbbbbbbbbbbbccccccccdd bbbbbbbbbbbbbbbbccccccccdd bbbbbbbbbbbbbbbbccccccccdd eeegggggggggggggccccccccdd eeegggggggggggggccccccccdd eeegggggggggggggccccccccdd eeegggggggggggggccccccccdd eeehhhhiiiiiiiiiiiiiiiiidd eeehhhhiiiiiiiiiiiiiiiiidd eeehhhhiiiiiiiiiiiiiiiiidd eeehhhhjjjjjkkkkkkkkkkkkdd eeehhhhjjjjjkkkkkkkkkkkkdd eeehhhhjjjjjkkkkkkkkkkkkdd eeehhhhjjjjjkkkkkkkkkkkkdd eeehhhhjjjjjlllllllmmmmmdd eeehhhhjjjjjlllllllmmmmmdd eeehhhhjjjjjlllllllmmmmmdd eeehhhhjjjjjlllllllmmmmmdd eeehhhhjjjjjlllllllmmmmmdd eeehhhhjjjjjlllllllmmmmmdd eeehhhhjjjjjlllllllmmmmmdd fffffffffffffffffffmmmmmdd fffffffffffffffffffmmmmmdd fffffffffffffffffffmmmmmdd Code:
aaaaaaaaaaaaaaaaaaaaaaaabbb aaaaaaaaaaaaaaaaaaaaaaaabbb ccdddddddddddddddeeeefffbbb ccdddddddddddddddeeeefffbbb ccdddddddddddddddeeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccgghhhhhhiiiiiiieeeefffbbb ccggjjjjjjlllllllllllfffbbb ccggjjjjjjlllllllllllfffbbb ccggjjjjjjlllllllllllfffbbb ccggjjjjjjlllllllllllfffbbb ccggjjjjjjmmmmmmmmmmmmnnnnn ccggjjjjjjmmmmmmmmmmmmnnnnn ccggjjjjjjmmmmmmmmmmmmnnnnn ccggjjjjjjmmmmmmmmmmmmnnnnn ccggkkkkkkkkoooooooooonnnnn ccggkkkkkkkkoooooooooonnnnn ccggkkkkkkkkoooooooooonnnnn ccggkkkkkkkkoooooooooonnnnn ccggkkkkkkkkoooooooooonnnnn ccggppppppppppppppppppppppp ccggppppppppppppppppppppppp Code:
aaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaa bbbbbbbbbbbbbbbbbbbbbbbbbccc bbbbbbbbbbbbbbbbbbbbbbbbbccc ddddddddddddddddddeeeffffccc ddddddddddddddddddeeeffffccc ddddddddddddddddddeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhjjjjjjkkkkkkkeeeffffccc hhhhhllllmmmmmmmmmeeeffffccc hhhhhllllmmmmmmmmmeeeffffccc iiiiillllmmmmmmmmmeeeffffccc iiiiillllmmmmmmmmmeeeggggggg iiiiillllmmmmmmmmmeeeggggggg iiiiillllmmmmmmmmmeeeggggggg iiiiillllnnnnnnnnnnnnggggggg iiiiillllnnnnnnnnnnnnggggggg iiiiillllnnnnnnnnnnnnggggggg iiiiillllnnnnnnnnnnnnggggggg iiiiillllooooooooooooooooooo iiiiillllooooooooooooooooooo iiiiillllooooooooooooooooooo 
20161129, 01:51  #8 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
you can upper bound if for even n by using either 2n if the solution for n/2 is greater than n\4 or 4 times the solution for n\2 otherwise. edit: so for example using the 17 by 17 bound in the video is 8 so we can say with confidence that for 34 by 34 the minimum solution is upper bounded by 32. and for the n=23 example above we can say that now for n=46 the minimal solution is not greater than 32 as well. sorry adding trivialities to the thread.
Last fiddled with by science_man_88 on 20161129 at 01:54 
20161129, 04:52  #9  
Aug 2012
Mass., USA
100111110_{2} Posts 
Quote:
A little bit more about my initial find... This web page listed what had been believed to be the optimal Mondrian scores for 4x4 up to 17x17. So after I found solutions that I thought were optimal for squares up to 17x17, I looked for a solution for the 18x18 square with a score that matched the 17x17 value. After my program generated many solutions (having a score of 8), I tried to find one manually on my own. I struggled for awhile, so I finally peeked at part of one of the solutions found by my computer. I was able to complete the 18x18 square by trial and error from the set of rectangles the computer was using, and that is the solution I posted. It probably wasn't until a couple days later or so that I realized that this solution had a score that was better by 2 than what was being claimed to be the best possible score. Until then, I didn't think my solution was anything special. Of course, I then realized I should post my solution online right away. 

20161129, 19:44  #10 
Nov 2016
2^{4} Posts 
Many thanks for the improved solutions. I found a few bugs in my coding. I'll be posting an update at http://demonstrations.wolfram.com/MondrianArtProblem/ in the near future.

20161130, 21:10  #11  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10010011001011_{2} Posts 
Quote:


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