References for Rectifiable Polyominoes
These are references for tiling rectangles with copies of a single polyomino, tiling rectangles with copies of two types of a polyominoes, 3-dimensional (and higher) analogues, and also some closely related topics. Links to Mathematical Reviews (MR) require a subscription. Other links may also be behind paywalls.
• James Bitner, Tiling 5n × 12 Rectangles with Y-pentominoes, Journal of Recreational Mathematics 7 (1974), no. 4, pp. 276-278. [MR]
• Olivier Bodini, Tiling a Rectangle with Polyominoes, Discrete Models for Complex Systems (DMCS'03), pp. 81-88. [MR]
• Maarten Bos, Tiling Squares with Two Different Hexominoes, Cubism For Fun 70 (July 2007), pp. 4-7.
• C.J. Bouwkamp and D.A. Klarner, Packing a Box with Y-pentacubes, Journal of Recreational Mathematics 3 (1970), no. 1, pp. 10-26.
• Chris Bouwkamp, The Cube-Y Problem, Cubism For Fun 25 (December 1990 - January 1991), part 3, pp. 30-43.
• Andrejs Cibulis and Ilvars Mizniks, Tiling Rectangles with Pentominoes, Latvijas Universitātes Zinātniskie Raksti 612 (1998) pp. 57-61.
• Andris Cibulis, Packing Boxes with N-tetracubes, Crux Mathematicorum with Mathematical Mayhem 23 (October 1997), no. 6, pp. 336-342.
• Andris Cibulis and Andy Liu, Packing Rectangles with the L and P Pentominoes, Math Horizons 9 (November 2001), no. 2, pp. 30-31.
• Andrew L. Clarke, A Pentomino Conjecture, Problem 600, Journal of Recreational Mathematics 10 (1977-78), no. 1, p. 54.
◦ Solution by Mike Beeler, Journal of Recreational Mathematics 12 (1979-80), no. 1, pp. 63-64.
• Andrew L. Clarke, Packing Boxes with Congruent Polycubes, Journal of Recreational Mathematics 10 (1977-78), no. 3, pp. 177-182.
• Karl A. Dahlke, The Y-hexomino has order 92, Journal of Combinatorial Theory, Series A 51 (1989), no. 1, pp. 125-126. [MR]
• Karl A. Dahlke, A Heptomino of Order 76, Journal of Combinatorial Theory, Series A 51 (1989), no. 1, pp. 127-128. [MR]
◦ Erratum, Journal of Combinatorial Theory, Series A 52 (1990), no. 2, p. 321. [MR]
• Karl A. Dahlke, Solomon W. Golomb and Herbert Taylor, An Octomino of High Order, Journal of Combinatorial Theory, Series A 70 (1995), no. 1, pp. 157-158. [MR]
• N.G. de Bruijn and D.A. Klarner, A finite basis theorem for packing boxes with bricks, Philips Research Reports 30 (1975), pp. 337-343.
• Raymond R. Fletcher III, Tiling Rectangles with Symmetric Hexagonal Polyominoes, Proceedings of the Twenty-seventh Southeastern International Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, 1996, Congressus Numerantium 122 (1996), pp. 3-29. [MR]
• Julian Fogel, Mark Goldenberg and Andy Liu, Packing Rectangles with Y-Pentominoes, Mathematics and Informatics Quarterly 11 (2001), no. 3, pp. 133-137.
• Martin Gardner, Polyominoes and Rectification, Chapter 13 in Mathematical Magic Show, The Mathematical Association of America, 1989.
• Frits Göbel, Packing with Congruent Shapes, Cubism For Fun 22 (December 1989), pp. 13-20.
• Frits Göbel, Prime pentacube packing, Cubism For Fun 33 (February 1994), pp. 24-25.
• S.W. Golomb, Covering a Rectangle with L-tetrominoes, Problem E 1543, American Mathematical Monthly 69 (November 1962), no. 9, p. 920.
◦ Solution to Problem E 1543, D.A. Klarner, American Mathematical Monthly 70 (August-September 1963), no. 7, pp. 760-761.
• Solomon W. Golomb, Tiling with Polyominoes, Journal of Combinatorial Theory 1 (1966) pp. 280-296. [MR]
• Solomon W. Golomb, Tiling with Sets of Polyominoes, Journal of Combinatorial Theory 9 (1970) pp. 60-71. [MR]
• Solomon W. Golomb, Polyominoes Which Tile Rectangles, Journal of Combinatorial Theory, Series A 51 (1989), no. 1, pp. 117-124. [MR]
• Solomon W. Golomb, Tiling Rectangles with Polyominoes, Chapter 8 in Polyominoes, Second edition, Princeton University Press, 1994.
• Solomon W. Golomb, Tiling Rectangles with Polyominoes, in Mathematical entertainments, edited by David Gale, The Mathematical Intelligencer 18 (1996), no. 2, pp. 38-47. [MR]
• Jenifer Haselgrove, Packing a Square with Y-pentominoes, Journal of Recreational Mathematics 7 (1974), no. 3, p. 229.
• Robert Hochberg and Michael Reid, Tiling with Notched Cubes, Discrete Mathematics 214 (2000), no. 1-3, pp. 255-261. [MR] [Zbl]
• Ross Honsberger, Box packing problems, chapter 8 in Mathematics Gems II, the Mathematical Association of America, Washington D.C. 1976.
◦ Ross Honsberger, Packungsprobleme, chapter 8 in Mathematische Juwelen, Springer Vieweg, 1982 (German translation of previous)
• Charles H. Jepsen, Lowell Vaughn and Daren Brantley, Orders of L-shaped Polyominoes, Journal of Recreational Mathematics 32 (2003-2004), no. 3, pp. 226-231.
• Michał Kieza, Zbudujmy z klocków prostopadłościan (Polish), Matematyka-Społeczeństwo-Nauczanie 46 (2011), pp. 32-40.
• David A. Klarner, Some Results Concerning Polyominoes, Fibonacci Quarterly 3 (1965), pp. 9-20. [MR]
• David A. Klarner, Packing a Rectangle with Congruent N-ominoes, Journal of Combinatorial Theory 7 (1969) pp. 107-115. [MR]
• David A. Klarner, Letter to the Editor, Journal of Recreational Mathematics 3 (1970), no. 4, p. 258.
• David A. Klarner, A Finite Basis Theorem Revisited, Technical Report CS-TR-73-338, Stanford University, February 1973.
• David Klarner, A Search for N-pentacube Prime Boxes, Journal of Recreational Mathematics 12 (1979-80), no. 4, pp. 252-257. [MR]
• D.A. Klarner and F. Göbel, Packing boxes with congruent figures, Indagationes Mathematicae 31 (1969) pp. 465-472. [MR]
• Earl S. Kramer, Tiling Rectangles with T and C Pentominoes, Journal of Recreational Mathematics 16 (1983-84), no. 2, pp. 102-113. [MR]
• Earl S. Kramer and Frits Göbel, Tiling Rectangles with Pairs of Pentominoes, Journal of Recreational Mathematics 16 (1983-84), no. 3, pp. 198-206. [MR]
• Rodolfo Marcelo Kurchan, Letter to the Editor, Journal of Recreational Mathematics 23 (1991), no. 1, p. 5.
• Rodolfo Marcelo Kurchan, Letter to the Editor, Journal of Recreational Mathematics 24 (1992), no. 3, pp. 184-185.
• Miklós Laczkovich, Tiling with T-tetrominoes, Problem 1263, Mathematics Magazine 60 (April 1987), no. 2, p. 114.
◦ Solution to Problem 1263, Jerrold W. Grossman, Mathematics Magazine 61 (April 1988), no. 2, pp. 119-120.
• Andy Liu, Packing Rectangles with Polynominoes, Mathematical Medley 30 (June 2003), no. 1, pp. 2-11.
• T.W. Marlow, Grid Dissections, Chessics 23 (1985), pp. 78-79.
• William Rex Marshall, Packing Rectangles with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 77 (1997), no. 2, pp. 181-192. [MR]
• Jean Meeus, The Smallest U-N Square, Journal of Recreational Mathematics 18 (1985-86), no. 1, p. 8.
• Jean Meeus, Letter to the Editor, Journal of Recreational Mathematics 18 (1985-86), no. 1, pp. 49, 51.
• Michael Reid, Letter to the Editor, Journal of Recreational Mathematics 25 (1993), no. 2, pp. 149-150.
• Michael Reid, Tiling Rectangles and Half Strips with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 80 (1997), no. 1, pp. 106-123. [MR] [Zbl]
• Michael Reid, Tiling a Square with Eight Congruent Polyominoes, Journal of Combinatorial Theory, Series A 83 (1998), no. 1, p. 158. [Zbl]
• Michael Reid, Tiling with Similar Polyominoes, Journal of Recreational Mathematics 31 (2002-2003), no. 1, pp. 15-24.
• Michael Reid, Tile Homotopy Groups, L'Enseignement Mathématique 49 (2003), no. 1-2, pp. 123-155. [MR] [Zbl]
• Michael Reid, Klarner Systems and Tiling Boxes with Polyominoes, Journal of Combinatorial Theory, Series A 111 (2005), no. 1, pp. 89-105. [MR] [Zbl]
• Michael Reid, Asymptotically Optimal Box Packing Theorems, The Electronic Journal of Combinatorics 15 (2008), no. 1, R78, 19 pp. [MR] [Zbl]
• Michael Reid, Many L-Shaped Polyominoes Have Odd Rectangular Packings, Annals of Combinatorics 18 (2014) pp. 341-357. [MR] [Zbl]
• Karl Scherer, Some New Results on Y-pentominoes, Journal of Recreational Mathematics 12 (1979-80), no. 3, pp. 201-204. [MR]
• Karl Scherer, Heptomino Tessellations, Problem 1045, Journal of Recreational Mathematics 14 (1981-82), no. 1, p. 64.
◦ Solutions by Scherer, and Karl A. Dahlke, Journal of Recreational Mathematics 21 (1989), no. 3, pp. 221-223.
◦ Solution by Karl A. Dahlke, Journal of Recreational Mathematics 22 (1990), no. 1, pp. 68-69.
• Karl Scherer, A Puzzling Journey To The Reptiles And Related Animals, privately published, Auckland, New Zealand, 1987.
• Karl Scherer, Pentacube Packing Problems, Problem 1615, Journal of Recreational Mathematics 20 (1988), no. 1, p. 78.
◦ Solution by Richard I. Hess, Journal of Recreational Mathematics 21 (1989), no. 1, pp. 74-75.
◦ Solution by Karl Scherer, Journal of Recreational Mathematics 24 (1992), no. 1, pp. 62-64.
• Karl Scherer, The U-Pentacube Packing Problem, Problem 1963, Journal of Recreational Mathematics 24 (1992), no. 2, p. 146.
◦ Solutions by Brian Barwell and Michael Reid, Journal of Recreational Mathematics 25 (1993), no. 3, pp. 226-229.
• Karl Scherer, The T-Pentacube Packing Problem, Problem 1990, Journal of Recreational Mathematics 24 (1992), no. 3, p. 224.
◦ Solutions by Frits Göbel and Michael Beeler, Journal of Recreational Mathematics 26 (1994), no. 1, pp. 66-67.
• Karl Scherer, The primes of a certain pentacube, Journal of Recreational Mathematics 26 (1994), no. 1, pp. 1-2.
• Robert Spira, A Pavement of Tetrominoes, Problem E 1786, American Mathematical Monthly 72 (May 1965), no. 5, p. 543.
◦ Solution to Problem E 1786, American Mathematical Monthly 73 (June-July 1966), no. 6, p. 673.
• Robert Spira, Impossibility of Covering a Rectangle with L-Hexominoes, Problem E 1983, American Mathematical Monthly 74 (April 1967), no. 4, p. 439.
◦ Solution to Problem E 1983, Dennis Gannon, American Mathematical Monthly 75 (August-September 1968), no. 7, pp. 785-786.
• I. N. Stewart and A. Wormstein, Polyominoes of Order 3 Do Not Exist, Journal of Combinatorial Theory, Series A 61 (September 1992), no. 1, pp. 130-136. [MR]
• Pieter Torbijn and Aad van der Wetering, Tiling Squares with Two Different Pentominoes, Cubism For Fun 68 (November 2005), pp. 16-17.
• Johan van de Konijnenberg, Finding Prime Boxes of Pentacubes, Cubism For Fun 79 (July 2009), pp. 18-20.
• D.W. Walkup, Covering a Rectangle with T-tetrominoes, American Mathematical Monthly 72 (November 1965), no. 9, pp. 986-988. [MR]
• Ingo Wrede, Rechteckzerlegungen mit kleinen Polyominos, Diplomarbeit, (1990) Technische Universität Braunschweig.
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Updated September 12, 2015.