A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viewed as the converse of Tarski’s plank problem (Bang’s theorem): Is it true that if the total width of a collection of planks is sufficiently large, then the planks can be individually translated to cover a unit ball B? A translative covering of B by planks is said to be non-dissective if the planks can be added one by one, in some order, such that the uncovered part remains connected at each step and is empty at the end. Improving a classical result of Groemer, we show that every set of C/ε^7/4 planks of width ε admits a non-dissective translative covering of a 3-dimensional ball B³, provided C is large enough. Our proof yields a low-complexity algorithm. We also show that c/ε^4/3 planks are, in general, insufficient for a non-dissective covering of B³. This provides the first non-trivial lower bound for this problem.
Kupavskii et al. (2026) studied this question.