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21
Tiling Rectangles And Half Strips With Congruent Polyominoes
, 1996
"... In this paper, we present three new polyominoes that tile rectangles, as well as a new family of polyominoes that tile rectangles. We also give three families of polyominoes, each of which tiles an infinite half strip. All previous examples of polyominoes that tile half strips were either already kn ..."
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In this paper, we present three new polyominoes that tile rectangles, as well as a new family of polyominoes that tile rectangles. We also give three families of polyominoes, each of which tiles an infinite half strip. All previous examples of polyominoes that tile half strips were either already known to tile a rectangle, or were later found to tile a rectangle. It is still unknown if every polyomino that tiles a half strip also tiles a rectangle.
When Can You Tile a Box with Translates of Two Given Rectangular Bricks?
 Electr. J. Combin
, 2004
"... When can a ddimensional rectangular box R be tiled by translates of two given ddimensional rectangular bricks B 1 and B 2 ?WeprovethatR can be tiled by translates of B 1 and B 2 if and only if R can be partitioned by a hyperplane into two subboxes R 1 and R 2 such that R i can be tiled by transl ..."
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When can a ddimensional rectangular box R be tiled by translates of two given ddimensional rectangular bricks B 1 and B 2 ?WeprovethatR can be tiled by translates of B 1 and B 2 if and only if R can be partitioned by a hyperplane into two subboxes 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 su#cient 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 d , and let B 1 and B 2 be of respective sizes v 1 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,...,d;or (b) z i /w i is an integer for i =1, 2,...,d;or (c) there is an index k such that z i /v i and z i /w i are integers for all i k, and z k = #v k + #w k for some nonnegative integers # and #.
KLARNER SYSTEMS AND TILING BOXES WITH POLYOMINOES
 JOURNAL OF COMBINATORIAL THEORY, SERIES A, 111 (2005), NO. 1, PP. 89–105.
, 2005
"... Let T be a protoset of ddimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by T? A nice result of Klarner and Göbel asserts that the answer to this question can always be given in a particularly simple form, namely, by giving a finite list of “prime ” boxes. All other ..."
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Let T be a protoset of ddimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by T? A nice result of Klarner and Göbel asserts that the answer to this question can always be given in a particularly simple form, namely, by giving a finite list of “prime ” boxes. All other boxes that can be tiled can be deduced from these prime boxes. We give a new, simpler proof of this fundamental result. We also show that there is no upper bound to the number of prime boxes, even when restricting attention to singleton protosets. In the last section, we determine the set of prime rectangles for several small polyominoes.
TILING WITH SIMILAR POLYOMINOES
 JOURNAL OF RECREATIONAL MATHEMATICS 31 (2002–2003), NO. 1, PP. 15–24.
, 2003
"... Numerous authors have studied tilings that use congruent copies of a single polyomino shape, for example ..."
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Numerous authors have studied tilings that use congruent copies of a single polyomino shape, for example
dugosija_konacno.PDF
, 2000
"... Abstract: Abstract: In this paper we found an upper bound on the number of items of the rectangular form b a× that can be loaded onto a rectangular pallet B A × , such that the sides of the loaded items are parallel to the sides of the pallet and the interiors of the loaded items do not overlap. ..."
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Abstract: Abstract: In this paper we found an upper bound on the number of items of the rectangular form b a× that can be loaded onto a rectangular pallet B A × , such that the sides of the loaded items are parallel to the sides of the pallet and the interiors of the loaded items do not overlap.
Pavage des Polyominos et Bases de Gröbner
, 2001
"... In this paper, we answer to a question of Grunbaum by proving that, for all set F of polyominoes (union of unit squares of a square lattice), we can nd a Ztiling (signed tile) of polyominoes by copies of elements of F in polynomial time. We use for this the theory of generalised Gröbner bases. ..."
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In this paper, we answer to a question of Grunbaum by proving that, for all set F of polyominoes (union of unit squares of a square lattice), we can nd a Ztiling (signed tile) of polyominoes by copies of elements of F in polynomial time. We use for this the theory of generalised Gröbner bases. For instance, we can algorithmicaly nd again and extend results of Lagarias and Romero on the topic.