Tarski's plank problem asks for the minimal number of planks (each 1 foot wide and length greater than 9 feet) needed to cover a circular table of diameter 9 feet. Nine parallel planks suffice, but is there a covering using fewer planks if suitably oriented?
For the circular-table case, surface area on a sphere induces a measure on its equatorial disk for which the measure covered by a plank is at most a constant times its width (King 1994).
It was solved by an ingenious argument of Bang (1950, 1951).
More generally, a plank in Euclidean space is the closed region between two distinct parallel hyperplanes,
and its width is their distance. Bang (1951) proved
that planks covering a convex body have total width
at least the minimum width of that body. Bakaev and Polyanskii (2026) extended this
type of bound to a unit cube 
with 
after removing the interior
of a box 
whose sides are parallel to those of 
. If the side lengths of 
satisfy 
, then every finite plank covering of
has total width at least 1, and the bound is sharp. The planar case gives a unit
square with an axis-parallel square hole of side length 
. Bakaev and Polyanskii (2026) state that ChatGPT
implemented their idea of applying Bang's lemma after an affine
transformation of the coordinates.
