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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.

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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 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.

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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 $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\$.

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