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.

See also

Cylinder Covering

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References

Aharoni, R.; Holzman, R.; Krivelevich, M.; and Meshulam, R. "Fractional Planks." Disc. Comput. Geom. 27, 585-602, 2002.Bakaev, E. and Polyanskii, A. "A Sharp Plank Theorem for Cube Annuli." 30 Oct 2026. https://arxiv.org/abs/2611.00379v1.Bang, T. "On Covering by Parallel-Strips." Mat. Tidsskr. B, 49-53, 1950.Bang, T. "A Solution to the 'Plank Problem.' " Proc. Amer. Math. Soc. 2, 990-993, 1951.King, J. L. "Three Problems in Search of a Measure." Amer. Math. Monthly 101, 609-628, 1994.Tarski, A. "Further Remarks about the Degree of Equivalence of Polygons." [Polish]. Odbitka Z. Parametru. 2, 310-314, 1932.

Referenced on Wolfram|Alpha

Tarski's Plank Problem

Cite this as:

Weisstein, Eric W. "Tarski's Plank Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TarskisPlankProblem.html

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