An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more nonintersecting sets so that the oblongs in any of these sets tile a square?
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1$\begingroup$ It seems that searching the first 25 oblongs tiling squares with sidelength 60 and 40 may be promising. $\endgroup$ – Bullet51 Oct 31 '18 at 3:22

$\begingroup$ Do you have any examples of oblongs tiling a square, Bernardo? $\endgroup$ – Gerry Myerson Oct 31 '18 at 12:51

$\begingroup$ A converse problem of tiling oblongs with squares was posed at math.stackexchange.com/questions/2057290/… $\endgroup$ – Gerry Myerson Oct 31 '18 at 12:57

2$\begingroup$ Yes, @GerryMyerson: Tiles 2x3, 3x4, 4x5, 5x6, 6x7, 7x8, and 9x10 tile a square of side 16. I think they are pretty common. $\endgroup$ – Bernardo Recamán Santos Oct 31 '18 at 13:07