So, a simple perfect squaring of a rectangle is a tiling of that rectangle by squares whose side lengths are all distinct integers. Additionally, not subset of the squares must form a smaller rectangle. My question is if there is a simple perfect squaring of a 1366 by 768 rectangle?
We could try reducing it to a simpler problem by splitting the rectangle into a square and a smaller rectangle, but then we need to ensure that the side lengths are different, and that the their combination is simple. So we basically back to where we started.
P.S. If you are wondering why 1366 by 768, that's the dimension of my monitor, which I am trying to artistically square (hence the art tag).
I looked through some of the data on Squaring.Net (actually, copies at the Internet Archive, since the site is currently down), and there is no $1366\times768$ simple perfect squared rectangle of order $17$ or less; there may be one of higher order, but that's harder to check. But there is a $1354\times764$ simple perfect squared rectangle of order $17$, so if you can accept a six-pixel border on the left and right and a two-pixel border at the top and bottom, this will do the job. It looks like this:
The Bouwkamp code for this rectangle dissection is $17\ 1354\ 764\ (389, 403, 311, 251)\ (60, 191)\ (83, 157, 131)\ (375, 14)\ (9, 74)\ (361, 65)\ (322)\ (296)$.
EDIT: As Servaes points out, I overlooked an even better $1366\times766$ alternative, also of order $17$:
Its Bouwkamp code is $17\ 1366\ 766\ (419, 288, 258, 401) (115, 143) (203, 85) (200) (179, 365) (347, 72) (275) (193, 7) (186)$.