Tarski's Planks 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:

^{ (Snapshot from Bang's paper.) }

Does anyone know if this problem has been solved? I am especially interested in $\mathbb{R}^2$.

Bang, Thøger (1951), "A solution of the 'plank problem'",

Proc. Amer. Math. Soc.2 (6): 990–993, doi:10.2307/2031721, JSTOR 2031721, MR 0046672.