Minkowski's conjecture states that every lattice tiling of by unit hypercubes contains two hypercubes that meet in
face. Minkowski first considered the problem in 1896, when he stated it as a theorem
whose proof would be provided later. However, it subsequently appeared as on open
problem in Minkowski's 1907 book, suggesting the purported proof was erroneous. The
conjecture was subsequently proved in eight and fewer dimensions by Peron (1940)
and in general by Hajós (1942).
Keller's conjecture is a generalization of Minkowski's conjecture (which however is known to be true only in dimensions six
and less and to be false in at least dimensions 8, 10, and 12).
Debroni, J.; Eblen, J. D.; Langston, M. A.; Shor, P.; Myrvold, W.; and Weerapurage, D. "A Complete Resolution of the Keller Maximum
Clique Problem." Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms.
pp. 129-135, 2011. http://www.siam.org/proceedings/soda/2011/SODA11_011_debronij.pdf.Hajós,
G. "Über einfache und mehrfache Bedeckung des -dimensionalen Raumes mit einen Würfelgitter." Math.
Z.47, 427-467, 1942.Mackey, J. "A Cube Tiling of Dimension
Eight with No Facesharing." Disc. Comput. Geom.28, 275-279, 2002.Minkowski,
H. Diophantische Approximationen. Leipzig, Germany: Teubner, 1907.Peron,
O. "Über lückenlose Ausfüllung des -dimensionalen raumes durch kongruente Würfel I &
II." Math. Z.46, 1-26 and 161-180, 1940.