{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T08:47:36Z","timestamp":1771922856109,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>When can a $d$-dimensional rectangular box $R$ be tiled by translates of two given $d$-dimensional rectangular bricks $B_1$ and $B_2$? We prove that $R$ can be tiled by translates of $B_1$ and $B_2$ if and only if $R$ can be partitioned by a hyperplane into two sub-boxes $R_1$ and $R_2$ such that $R_i$ can be tiled by translates of the brick $B_i$ alone $(i=1,2).$ Thus an obvious sufficient condition for a tiling is also a necessary condition.  (However, there may be tilings that do not give rise to a bipartition of $R.$) There is an equivalent formulation in terms of the  (not necessarily integer) edge lengths of $R,$ $B_1,$ and $B_2.$ Let $R$ be of size $z_1\\times z_2\\times \\cdots\\times z_d,$ and let $B_1$ and $B_2$ be of respective sizes $v_1\\times v_2\\times \\cdots\\times v_d$ and $w_1\\times w_2\\times \\cdots\\times w_d.$ Then there is a tiling of the box $R$ with translates of the bricks $B_1$ and $B_2$ if and only if (a)   $z_i\/v_i$ is an integer for $i=1,2,\\ldots, d;$  or (b)   $z_i\/w_i$ is an integer for $i=1,2,\\ldots,d;$  or (c)   there is an index $k$ such that $z_i\/v_i$ and $z_i\/w_i$  are integers for all $i\\neq k,$ and $z_k=\\alpha v_k+\\beta w_k$ for some nonnegative integers $\\alpha$ and $\\beta.$ Our theorem extends some well known results (due to de Bruijn and Klarner) on tilings of rectangles by rectangles with integer edge lengths.<\/jats:p>","DOI":"10.37236\/1848","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:22:25Z","timestamp":1578720145000},"source":"Crossref","is-referenced-by-count":1,"title":["When Can You Tile a Box With Translates of Two Given Rectangular Bricks?"],"prefix":"10.37236","volume":"11","author":[{"given":"Richard J.","family":"Bower","sequence":"first","affiliation":[]},{"given":"T. S.","family":"Michael","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,5,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1n7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1n7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:02:52Z","timestamp":1579323772000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1n7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,5,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1848","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,5,14]]},"article-number":"N7"}}