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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?

larry01
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  • Strictly increasing? And you want to find such sequences? – amsmath Oct 08 '17 at 15:59
  • yes, increasing means strictly increasing; I ask if there exists at least one pair of such sequences – larry01 Oct 08 '17 at 16:21
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    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. – Ross Millikan Oct 08 '17 at 20:47
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    This was asked [here](https://math.stackexchange.com/questions/1164035/regular-way-to-fill-a-1-times1-square-with-frac1n-times-frac1n1-re) and no successful answer was given. – Ross Millikan Oct 08 '17 at 20:50
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    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. – Misha Lavrov Oct 08 '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.

Ross Millikan
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