https://hal.inria.fr/hal-01182962Meyerowitz, AaronAaronMeyerowitzFlorida Atlantic University [Boca Raton]Tiling the Line with TriplesHAL CCSD2001Tilingone dimensiondirect proof[INFO] Computer Science [cs][INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]Episciences Iam, CoordinationCori, Robert and Mazoyer, Jacques and Morvan, Michel and Mosseri, Rémy2015-08-06 10:44:492017-03-07 15:00:242015-08-06 11:24:00enConference papershttps://hal.inria.fr/hal-01182962/document10.46298/dmtcs.2282application/octet-steam1It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.