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.

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

Author

Ed Pegg Jr, Dec 15 2016

Keywords

• hard
• more
• nonn

Comments

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).

Examples

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

Links

• 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.

Crossrefs

Cf. A276523, A278970, A279848.

Extensions

Moved terms to A279848, expanded best values known

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