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Given a rectangular tile with dimensions $\frac{1}{n}\times\frac{1}{n+1}$ (and therefore area $\frac{1}{n(n+1)}$) for each positive integer $n$, can the $1\times 1$ square be tiled without gap or overlap using the tiles?

The sum of the areas of the tiles is exactly $1$, which can be demonstrated as follows:

$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\sum_{n=1}^{\infty} \frac{1}{n}-\frac{1}{n+1}=\Big(\frac{1}{1}-\frac{1}{2}\Big)+\Big(\frac{1}{2}-\frac{1}{3}\Big)+\Big(\frac{1}{3}-\frac{1}{4}\Big)+...=1$$

Thus it might seem plausible that a $1\times 1$ square could be tiled using tiles of the given dimensions (of course, not certain- some potentially unavoidable concave shape (for example) could possibly fail to admit any further squares despite nominally having the area to do so). However, to my knowledge (and some despite the aid of some brief googling), it is unknown whether such a tiling has been proven to exist, and if so, whether a procedure for placing the tiles has been described. A math teacher I currently work for (approximately the best person I know at math) claims that as of about ten years ago, the problem was not yet solved. So I wanted to ask if anyone knows whether the problem has a solution, and if so, what it is. Any pointers would be greatly appreciated.

Lieutenant Zipp
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  • A duplicate question... – GEdgar Jul 14 '22 at 21:07
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    Here it is: https://math.stackexchange.com/questions/1164035/ but it has been closed... I guess known open problems cannot be posted here. Anyway, see references there. – GEdgar Jul 14 '22 at 21:16
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    Lieutenant, your profile mentions quadratic reciprocity. See if you can prove that every even number is the sum of three pronic numbers ( if needed, we say $0$ is also pronic) – Will Jagy Jul 14 '22 at 22:03
  • @GEdgar Okay, thank you very much! – Lieutenant Zipp Jul 14 '22 at 22:10
  • @WillJagy Oooh, interesting. Will do! – Lieutenant Zipp Jul 14 '22 at 22:10
  • I’m voting to close this question because it is a research-level unsolved question. – Lieutenant Zipp Jul 14 '22 at 22:12
  • @WillJagy ...wait a minute, this is just $2*$Gauss' "EYPHKA" theorem. Still fun, though. – Lieutenant Zipp Jul 14 '22 at 22:16
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    Lieu, possible suggestion: I know of two books that combine quadratic forms (well, binary) and quadratic fields, building up a portion of the vocabulary of algebraic number theory in this setting. Suitable for self study. One is Lehman https://bookstore.ams.org/dol-52/ and I just got Trifkovic https://link.springer.com/book/10.1007/978-1-4614-7717-4 – Will Jagy Jul 15 '22 at 00:34

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