[Tarski's Planks problem](https://en.wikipedia.org/wiki/Tarski%27s_plank_problem),
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a disk of diameter $d$.
The last paragraph of Bang's short paper poses an open problem:
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[![Bang][1]][1]
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<sup>
(Snapshot from Bang's paper.)
</sup>
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Does anyone know if this problem has been solved?
I am especially interested in $\mathbb{R}^2$.

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> Bang, Thøger (1951), "[A solution of the 'plank problem'](https://doi.org/10.1090/S0002-9939-1951-0046672-4 )", *Proc. Amer. Math. Soc.* 2 (6): 990–993, doi:10.2307/2031721, [JSTOR 2031721](http://www.jstor.org/stable/2031721), MR 0046672.


  [1]: https://i.sstatic.net/CJBxg.png