{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:03:57Z","timestamp":1760144637134,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T00:00:00Z","timestamp":1715126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Let Vn(d) denote the least number, such that every collection of n\u00a0d-cubes with total volume 1 in d-dimensional (Euclidean) space can be packed parallelly into some d-box of volume Vn(d). We show that V3(d)=r1\u2212dd if d\u226511 and V3(d)=1r+1rd+1r\u2212rd+1 if 2\u2264d\u226410, where r is the only solution of the equation 2(d\u22121)kd+dkd\u22121=1 on 22,1 and (k+1)d(1\u2212k)d\u22121dk2+d+k\u22121=kddkd+1+dkd+kd+1 on 22,1, respectively. The maximum volume is achieved by hypercubes with edges x, y, z, such that x=2rd+1\u22121\/d, y=z=rx if d\u226511, and x=rd+(1r\u2212r)d+1\u22121\/d, y=rx, z=(1r\u2212r)x if 2\u2264d\u226410. We also proved that only for dimensions less than 11 are there two different maximum packings, and for all dimensions greater than 10, the maximum packing has the same two smallest cubes.<\/jats:p>","DOI":"10.3390\/a17050198","type":"journal-article","created":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T09:58:56Z","timestamp":1715162336000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Three Cube Packing for All Dimensions"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6256-4475","authenticated-orcid":false,"given":"Peter","family":"Adamko","sequence":"first","affiliation":[{"name":"Department of Quantitative Methods and Economics Informatics, Faculty of Operation and Economics of Transport and Communications, University of \u017dilina, Univerzitn\u00e1 1, 010 26 \u017dilina, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0166-218X(91)90071-4","article-title":"Problems, problems, problems","volume":"31","author":"Moser","year":"1991","journal-title":"Discret. Appl. Math."},{"key":"ref_2","unstructured":"Brass, P., Moser, W.O.J., and Pach, J. (2005). Research Problems in Discrete Geometry, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Croft, H.T., Falconer, K.J., and Guy, R.K. (1991). Unsolved Problems in Geometry, Springer.","DOI":"10.1007\/978-1-4612-0963-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"103","DOI":"10.4064\/cm-17-1-103-110","article-title":"Some packing and covering theorems","volume":"17","author":"Moon","year":"1967","journal-title":"Colloq. Math."},{"key":"ref_5","unstructured":"Moser, W., and Pach, J. (1994). Research Problems in Discrete Geometry, McGill University."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1111\/j.1749-6632.1970.tb56476.x","article-title":"Packing squares in rectangles I","volume":"175","author":"Kleitman","year":"1970","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kleitman, D.J., and Krieger, M.M. (1975, January 13\u201315). An optimal bound for two dimensional bin packing. Proceedings of the 16th Annual Symposium on Foundations of Computer Science, Berkeley, CA, USA.","DOI":"10.1109\/SFCS.1975.6"},{"key":"ref_8","first-page":"35","article-title":"A note on a packing of squares","volume":"10","year":"1995","journal-title":"Stud. Univ. Transp. Commun. \u017dilina Math.-Phys. Ser."},{"key":"ref_9","first-page":"199","article-title":"On packing of four and five squares into a rectangle","volume":"19","year":"1999","journal-title":"Note Mat."},{"key":"ref_10","unstructured":"Novotn\u00fd, P. (2002, January 9\u201311). Vyu\u017eitie po\u010d\u00edta\u010da pri rie\u0161en\u00ed ukladacieho probl\u00e9mu. Proceedings of the Symposium on Computational Geometry, Ko\u010dovce, Slovakia. (In Slovak)."},{"key":"ref_11","unstructured":"Platz, A. (2016). A Proof of Moser\u2019s Square Packing Problem for Small Instances. [Master\u2019s Thesis, Universit\u00e4t Bonn, Forschungsinstitut f\u00fcr Diskrete Mathematik]."},{"key":"ref_12","first-page":"75","article-title":"On packing of squares into a rectangle","volume":"32","year":"1996","journal-title":"Arch. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1016\/j.comgeo.2011.05.001","article-title":"On packing squares into a rectangle","volume":"44","author":"Hougardy","year":"2011","journal-title":"Comput. Geom."},{"key":"ref_14","unstructured":"Ilhan, A. (2014). Das Packen von Quadraten in ein Rechteck. [Diploma Thesis, Universit\u00e4t Bonn, Forschungsinstitut f\u00fcr Diskrete Mathematik]."},{"key":"ref_15","unstructured":"Neuwohner, M. (2021). Reducing Moser\u2019s Square Packing Problem to a Bounded Number of Squares. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/S0021-9800(68)80047-X","article-title":"On packing of squares and cubes","volume":"5","author":"Meir","year":"1968","journal-title":"J. Comb. Theory"},{"key":"ref_17","unstructured":"Novotn\u00fd, P. (2011, January 19\u201321). Ukladanie kociek do kv\u00e1dra. Proceedings of the Symposium on Computational Geometry, Ko\u010dovce, Slovakia. (In Slovak)."},{"key":"ref_18","unstructured":"Novotn\u00fd, P. (2006, January 27\u201329). Pakovanie troch kociek. Proceedings of the Symposium on Computational Geometry, Ko\u010dovce, Slovakia. (In Slovak)."},{"key":"ref_19","unstructured":"Novotn\u00fd, P. (2007, January 24\u201326). Najhor\u0161ie pakovate\u013en\u00e9 \u0161tyri kocky. Proceedings of the Symposium on Computational Geometry, Ko\u010dovce, Slovakia. (In Slovak)."},{"key":"ref_20","first-page":"5","article-title":"Minimaliz\u00e1cia objemu kv\u00e1dra pre ulo\u017eenie troch kociek v dimenzii 4","volume":"12","author":"Adamko","year":"2015","journal-title":"Slov. \u010casopis Pre Geom. Graf."},{"key":"ref_21","unstructured":"B\u00e1lint, V., and Adamko, P. (2016, January 2\u20134). Minimization of the parallelepiped for packing of three cubes in dimension 6. Proceedings of the Aplimat: 15th Conference on Applied Mathematics, Bratislava, Slovakia."},{"key":"ref_22","first-page":"245","article-title":"Packing Three Cubes in 8-Dimensional Space","volume":"22","year":"2018","journal-title":"J. Geom. Graph."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Sedlia\u010dkov\u00e1, Z., and Adamko, P. (2021). Packing Three Cubes in D-Dimensional Space. Mathematics, 9.","DOI":"10.3390\/math9172046"},{"key":"ref_24","first-page":"1","article-title":"Universal asymptotical results on packing of cubes","volume":"28","author":"Adamko","year":"2016","journal-title":"Stud. Univ. \u017dilina Math. Ser."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/5\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:41:35Z","timestamp":1760107295000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/5\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,8]]},"references-count":24,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["a17050198"],"URL":"https:\/\/doi.org\/10.3390\/a17050198","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,5,8]]}}}