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Consider the set of rectangles $r_n | n \in \Bbb N$ such that rectangle $r_n$ has shape $\frac1n\times\frac1{n+1}$. The total area composed by one copy of each $r_n$ as $n$ ranges from $1$ to infinity is $1$. Call that set of rectangles $S$.

In Concrete Mathematics it is speculated that you can fit all the rectangles in $S$ into a unit square, without overlap of any interior area. It is also speculated (by the second author) that they can't all be fit, and it is presented as a research problem.

I have a computer-based approach which could potentially decide the issue if you can't fit them (but won't provide a proof if they can fit). In particular, it seems to indicate that if you add the restriction that the rectangles (ordered by area) are forced to alternate long-side-vertical and long-side-horizontal, you come to an impasse.

But since the volume I saw it in is 25 years old, I'm wondering, before implementing and error-checking and optimizing the method for arbitrary placement, whether this question has, in the intervening years, been resolved.

So my question is

Is it known whether $S$ can be packed into a unit square, without overlap?

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    $\begingroup$ Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, 1989. $\endgroup$ Mar 14, 2019 at 22:55
  • $\begingroup$ Duplicate of 34145. Gerhard "Going Link Light For Lent" Paseman, 2019.03.14. $\endgroup$ Mar 14, 2019 at 22:59
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    $\begingroup$ Unlike @GerhardPaseman, I'm not fasting: "Can we cover the unit square by these rectangles?": "The problem was still unsettled as of 2005." $\endgroup$ Mar 14, 2019 at 23:21

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