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This problem actually goes back to Leo Moser.

The best result that I'm aware of is due to D. Jennings, who proved that all the rectangles of size $k^{-1} × (k + 1)^{-1}$, $k = 1, 2, 3 ...$, can be packed into a square of size $(133/132)^2$ (link).

Edit 1. A web search via Google Scholar gave a reference to this ariclearticle by V. Bálint, which claims that the squares rectangles can be packed into a square of size $(501/500)^2$.

Edit 2. The state of art of this and related packing problems due to Leo Moser is discussed in Chapter 3 of "Research Problems in Discrete Geometry" by P.Brass, W. O. J. Moser and J. Pach. The problem was still unsettled as of 2005.

show/hide this revision's text 3 added 537 characters in body; deleted 1 characters in body

This problem actually goes back to Leo Moser.

The best result that I'm aware of is due to D. Jennings, who proved that all the rectangles of size $k^{-1} × (k + 1)^{-1}$, $k = 1, 2, 3 ...$, can be packed into a square of size $(133/132)^2$ (link).

Edit 1. A web search via Google Scholar gave a reference to this aricle by V. Bálint, which claims that the squares can be packed into a square of size $(501/500)^2$.

Edit 2. The state of art of this and related packing problems due to Leo Moser is discussed in Chapter 3 of "Research Problems in Discrete Geometry" by P.Brass, W. O. J. Moser and J. Pach. The problem was still unsettled as of 2005.

show/hide this revision's text 2 added 500 characters in body

This problem actually goes back to Leo Moser .

The best result that I'm aware of is due to D. Jennings, who proved that all the rectangles of size $1/k k^{-1} × 1/(k (k + 1)$1)^{-1}$, $k = 1, 2, 3 ...$ ..$, can be packed into a square of side size $133/132$ (133/132)^2$ (link).

Edit. A web search via Google Scholar gave a reference to this aricle by V. Bálint, which claims that the squares can be packed into a square of size $(501/500)^2$.

show/hide this revision's text 1