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Is it possible to divide the unit square onto infinitely many rectangles of dimensions $\frac{1}{x_n}\times\frac{1}{y_n}$ where $(x_n),(y_n)$ are increasing sequences of integers?

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  • $\begingroup$ Strictly increasing? And you want to find such sequences? $\endgroup$ – amsmath Oct 8 '17 at 15:59
  • $\begingroup$ yes, increasing means strictly increasing; I ask if there exists at least one pair of such sequences $\endgroup$ – larry01 Oct 8 '17 at 16:21
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    $\begingroup$ I strongly suspect not. We would have to have $x_n=1,2,3,4,5,\ldots, y_n=2,3,4,5,6,\ldots$ because the total area for these is $1$ and any other pair of strictly increasing sequences will have an area less than $1$. I don't know a proof that you will always get stuck trying to fit them. $\endgroup$ – Ross Millikan Oct 8 '17 at 20:47
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    $\begingroup$ This was asked here and no successful answer was given. $\endgroup$ – Ross Millikan Oct 8 '17 at 20:50
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    $\begingroup$ It might be the case that the usual $\frac1k \times \frac1{k+1}$ packing problem has an affirmative answer, and yet this problem (in which all of them have to have the same orientation) does not. $\endgroup$ – Misha Lavrov Oct 8 '17 at 22:13
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This paper shows that all these rectangles can be packed into a square of side $133/132$ and states that the question of whether they can be packed into the unit square is open.

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