

A276523


Partition an n X n square into multiple noncongruent integersided rectangles. a(n) is the least possible difference between the largest and smallest area.


4



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
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OFFSET

3,1


COMMENTS

Developed as the Mondrian Art Puzzle.
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(58)a(96) as follows: 16, 15, 18, 15, 16, 18, 15, 18, 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


LINKS



EXAMPLE

A size11 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 noncongruent rectangles of area 30 to 36:
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CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS

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(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


STATUS

approved



