A278970 -id:A278970 - cp's OEIS frontend

A276523 Partition an n X n square into multiple non-congruent integer-sided rectangles. a(n) is the least possible difference between the largest and smallest area.

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, 12, 13, 13, 12, 14, 12, 13, 14, 13, 14, 15, 14, 14, 15, 15, 14, 15, 15, 14, 15, 15, 15

Offset: 3

Ed Pegg Jr, Nov 15 2016

Developed as the Mondrian Art Puzzle.
The rectangles can be similar, though. - Daniel Forgues, Nov 22 2016
That is, there can be a 1x2 rectangle and a 2x4 rectangle (these are similar), but there can't be two 1x2 rectangles (these are congruent). - Michael B. Porter, Oct 13 2018
Upper bounds for a(n) are n if n is odd, and min(2*n, 4 * a(n/2)) if n is even. - Roderick MacPhee, Nov 28 2016
An upper bound seems to be ceiling(n/log(n))+3, or A050501+3. See A278970. Holds to at least a(96). - Ed Pegg Jr, Dec 02 2016
Best known values for a(66)-a(96) as follows: 16, 18, 19, 18, 19, 18, 20, 20, 20, 20, 19, 20, 21, 21, 20, 21, 20, 20, 21, 22, 18, 22, 20, 22, 24, 23, 22, 22, 24, 24, 24. - (shortened by Ruud H.G. van Tol, Oct 25 2024)

			A size-11 square can be divided into 3 X 4, 2 X 6, 2 X 7, 3 X 5, 4 X 4, 2 X 8, 2 X 9, and 3 X 6 rectangles. 18 - 12 = 6, the minimal area range.
The 14 X 14 square can be divided into non-congruent rectangles of area 30 to 36:
  aaaaaaaaaabbbb
  aaaaaaaaaabbbb
  aaaaaaaaaabbbb
  cccdddddddbbbb
  cccdddddddbbbb
  cccdddddddbbbb
  cccdddddddbbbb
  cccdddddddbbbb
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff
  ccceeeeeffffff

Michel Gaillard, Optimal tilings for n=58..65
Robert Gerbicz, Optimal tilings for n=3..57
Gordon Hamilton, Mondrian Art Puzzles (2015).
Gordon Hamilton and Brady Haran, Mondrian Puzzle, Numberphile video (2016)
Mersenneforum.org puzzles, Mondrian art puzzles
Cooper O'Kuhn, The Mondrian Puzzle: A Connection to Number Theory, arXiv:1810.04585 [math.CO], 2018.
Cooper O'Kuhn and Todd Fellman, The Mondrian Puzzle: A Bound Concerning the M(n)=0 Case, arXiv:2006.12547 [math.NT], 2020. See also Integers (2021) Vol. 21, #A37.
Ed Pegg Jr, Mondrian Art Problem.

Cf. A050501, A278970, A279596.

Bruce Norskog corrected a(18), and a recheck by Pegg corrected a(15) and a(19). - Charles R Greathouse IV, Nov 28 2016
Correction of a(14), a(16), a(23) and new terms a(25)-a(28) from Robert Gerbicz, Nov 28 2016
a(29)-a(44) from Robert Gerbicz, Dec 02 2016
a(45)-a(47) from Robert Gerbicz added, as well as best known values to a(96).
Correction of a(45), a(46) and new terms a(48)-a(57) from Robert Gerbicz, Dec 27 2016
a(58)-a(65) from Michel Gaillard, Oct 23 2020

A279596 Partition an n X n square into multiple integer-sided rectangles where no one is a translation of any other; a(n) is the least possible difference between the largest and smallest area.

Original entry on oeis.org

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, 7, 7, 8, 7, 6, 6, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9

Offset: 3

Ed Pegg Jr, Dec 15 2016

Similar to the Mondrian Art sequence (A276523), but allowing repetition of rectangles with different orientations.
Proved optimal to a(45) by R. Gerbicz.  Best values known for a(46)-a(96): 10, 12, 11, 12, 12, 8, 12, 12, 13, 12, 12, 14, 14, 15, 12, 15, 14, 15, 14, 16, 16, 15, 16, 16, 16, 17, 16, 17, 14, 17, 18, 16, 18, 16, 18, 15, 16, 18, 18, 16, 18, 17, 19, 20, 17, 17, 21, 20, 20, 21, 22.
Seems to be bounded above by ceiling(n/log(n)). The currently verified distances from this bound are 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 (A279848).

			The 9 X 9 square can be divided into non-translatable rectangles with
aaaaaaaab
ddddddeeb
fggghheeb
fggghheeb
fiiihheeb
fiiijjjjb
fiiijjjjb
fkkkkkkkb
ccccccccc

Robert Gerbicz, Optimal tilings for n = 3..45
Mersenneforum.org puzzles, Mondrian art puzzles.
Ed Pegg Jr, Mondrian Art Problem.
Ed Pegg Jr, Mondrian Art Problem Upper Bound for defect.

Cf. A276523, A278970, A279848.

Moved terms to A279848, expanded best values known

a(28)-a(45) from Robert Gerbicz, Jan 01 2017

A279848 Partition an n X n square into multiple integer-sided rectangles where no one is a translation of any other. a(n) is ceiling(n/log(n)) - the least possible difference between the largest and smallest area.

Original entry on oeis.org

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, 2, 2, 1, 3, 4, 4, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3

Offset: 3

Ed Pegg Jr, Dec 21 2016

If ceiling(n/log(n)) is an upper bound for the Mondrian Art Problem variant (A279596), a(n) is the amount by which the optimal value beats the upper bound.
Terms a(3) to a(45) verified optimal by R. Gerbicz.
Term a(103) is at least 9, defect 14 (630-616) with 17 rectangles.
Best values known for a(46) to a(96): 3, 1, 2, 1, 1, 5, 2, 2, 1, 2, 2, 1, 1, 0, 3, 0, 2, 1, 2, 0, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 0, 2, 0, 3, 1, 4, 3, 1, 1, 4, 2, 3, 1, 0, 4, 4, 0, 1, 1, 0, 0.

Mersenneforum.org puzzles, Mondrian art puzzles.
Ed Pegg Jr, Mondrian Art Problem.
Ed Pegg Jr, Mondrian Art Problem Upper Bound for defect.

Cf. A278970, A276523, A279596.

a(28)-a(45) from Robert Gerbicz, Jan 01 2017

cp's OEIS Frontend

A276523 Partition an n X n square into multiple non-congruent integer-sided rectangles. a(n) is the least possible difference between the largest and smallest area.

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A279596 Partition an n X n square into multiple integer-sided rectangles where no one is a translation of any other; a(n) is the least possible difference between the largest and smallest area.

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A279848 Partition an n X n square into multiple integer-sided rectangles where no one is a translation of any other. a(n) is ceiling(n/log(n)) - the least possible difference between the largest and smallest area.

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