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155:, partition the board into a quarter-board of size 2 × 2 that contains the removed square, and a large tromino formed by the other three quarter-boards. The tromino can be recursively dissected into unit trominoes, and a dissection of the quarter-board with one square removed follows by the induction hypothesis. In contrast, when a chessboard of this size has one square removed, it is not always possible to cover the remaining squares by I-trominoes.
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used this tiling as the basis for what has become known as Golomb's tromino theorem: if any square is removed from a 2 × 2 chessboard, the remaining board can be completely covered with L-trominoes. To prove this by
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trominoes (trominoes with reflections considered distinct). When rotations are also considered distinct, there are six
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trominoes: two I and four L shapes. They can be obtained by rotating the above forms by 90°, 180° and 270°.
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127:. Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an
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are not considered to be distinct shapes, there are only two different
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288:, Providence, RI: American Mathematical Society, pp. 205–217,
203:(2nd ed.). Princeton, New Jersey: Princeton University Press.
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This article is about the geometric shape. For the game similar to
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trominoes: "I" and "L" (the "L" shape is also called "V").
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119:smaller trominos of the same type, for any integer
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284:Nițică, Viorel (2003), "Rep-tiles revisited",
111:Geometrical dissection of an L-tromino (rep-4)
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396:Golomb's inductive proof of a tromino theorem
131:. In this context, the L-tromino is called a
115:Both types of tromino can be dissected into
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354:(1954). "Checker boards and polyominoes".
251:"Counting polyominoes: yet another attack"
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16:Geometric shape formed from three squares
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103:Rep-tiling and Golomb's tromino theorem
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123: > 1. That is, they are
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307:Robinson, E. Arthur Jr. (1999).
87:Since both free trominoes have
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357:American Mathematical Monthly
328:10.1016/S0019-3577(00)87911-2
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91:, they are also the only two
309:"On the table and the chair"
270:10.1016/0012-365X(81)90237-5
249:Redelmeier, D. Hugh (1981).
144:mutilated chessboard problem
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411:Interactive Tromino Puzzle
58:made of three equal-sized
34:All possible free trominos
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314:Indagationes Mathematicae
164:Previous and next orders
66:Symmetry and enumeration
62:connected edge-to-edge.
153:mathematical induction
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50:of size 3, that is, a
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256:Discrete Mathematics
89:reflection symmetry
226:Weisstein, Eric W.
195:Golomb, Solomon W.
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516:Higher dimensions
148:Solomon W. Golomb
142:Motivated by the
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564:Pseudo-polyomino
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129:aperiodic tiling
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405:Tromino Puzzle
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390:External links
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321:(4): 581–599.
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286:MASS selecta
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137:chair tiling
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641:WikiProject
549:Polydrafter
523:Polyominoid
459:Polyominoes
364:: 675–682.
263:: 191–203.
200:Polyominoes
76:reflections
601:Snake cube
559:Polyiamond
229:"Triomino"
181:References
25:Triominoes
670:Polyforms
596:Soma cube
569:Polystick
544:Polyabolo
492:Heptomino
482:Pentomino
477:Tetromino
451:Polyforms
234:MathWorld
175:Tetromino
125:rep-tiles
93:one-sided
72:rotations
48:polyomino
664:Category
611:Hexastix
528:Polycube
507:Decomino
502:Nonomino
497:Octomino
487:Hexomino
197:(1994).
159:See also
44:triomino
21:dominoes
617:Tantrix
606:Tangram
583:puzzles
554:Polyhex
472:Tromino
378:0067055
337:1820555
294:2027179
60:squares
54:in the
52:polygon
40:tromino
651:Portal
624:Tetris
591:Blokus
537:Others
467:Domino
376:
335:
292:
207:
170:Domino
23:, see
579:Games
133:chair
97:fixed
70:When
56:plane
46:is a
581:and
205:ISBN
81:free
74:and
413:at
398:at
366:doi
323:doi
265:doi
42:or
666::
374:MR
372:.
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333:MR
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290:MR
261:36
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38:A
443:e
436:t
429:v
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267::
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121:n
117:n
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
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