Sources page biographical material

6.F. POLYOMINOES, ETC.
See S&B, pp. 15 18. See 6.F.1, 6.F.3, 6.F.4 & 6.F.5 for early occurrences of polyominoes. See Lammertink, 1996 & 1997 for many examples in two and three dimensions.

NOTATION. Each of the types of puzzle considered has a basic unit and pieces are formed from a number of these units joined edge to edge. The notation N: n₁, n₂, .... denotes a puzzle with N pieces, of which n_i pieces consist of i basic units. If n_i are single digit numbers, the intervening commas and spaces will be omitted, but the digits will be grouped by fives, e.g. 15: 00382 11.

Polyiamonds: Scrutchin; John Bull; Daily Sketch; Daily Mirror; B. T.s Zig-Zag; Daily Mail; Miller (1960); Guy (1960); Reeve & Tyrrell; O'Beirne (2 & 9 Nov 1961); Gardner (Dec 1964 & Jul 1965); Torbijn; Meeus; Gardner (Aug 1975); Guy (1996, 1999); Knuth,

Polycubes: Rawlings (1939); Editor (1948); Niemann (1948); French (1948); Editor (1948); Niemann (1948); Gardner (1958); Besley (1962); Gardner (1972)

Solid Pentominoes: Nixon (1948); Niemann (1948); Gardner (1958); Miller (1960); Bouwkamp (1967, 1969, 1978); Nelson (2002);

Cylindrical Pentominoes: Yoshigahara (1992);

Polyaboloes: Hooper (1774); Book of 500 Puzzles (1859); J. M. Lester (1919); O'Beirne (21 Dec 61 &  18 Jan 62)

Polyhexes: Gardner (1967); Te Riele & Winter

Polysticks: Benjamin; Barwell; General Symmetrics; Wiezorke & Haubrich; Knuth; Jelliss;

Polyrhombs or Rhombiominoes: Lancaster (1918); Jones (1992).

Polylambdas: Roothart.

Polyspheres -- see Section 6.AZ.

GENERAL REFERENCES
G. P. Jelliss. Special Issue on Chessboard Dissections. Chessics 28 (Winter 1986) 137 152. Discusses many problems and early work in Fairy Chess Review.

Branko Grünbaum & Geoffrey C. Shephard. Tilings and Patterns. Freeman, 1987. Section 9.4: Polyiamonds, polyominoes and polyhexes, pp. 497-511. Good outline of the field with a number of references otherwise unknown.

Michael Keller. A polyform timeline. World Game Review 9 (Dec 1989) 4-5. This outlines the history of polyominoes and other polyshapes. Keller and others refer to polyaboloes as polytans.

Rodolfo Marcelo Kurchan (Parana 960 5 "A", 1017 Buenos Aires, Argentina). Puzzle Fun, starting with No. 1 (Oct 1994). This is a magazine entirely devoted to polyomino and other polyform puzzles. Many of the classic problems are extended in many ways here. In No. 6 (Aug 1995) he presents a labelling of the 12 hexiamonds by the letters A, C, H, I, J, M, O, P, S, V, X, Y, which he obtained from Anton Hanegraaf. I have never seen this before.

Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Vol. 1, recreation 23, pp. 64-66. Considers figures formed of isosceles right triangles. He has eight of these, coloured with eight colours, and uses some of them to form "chequers or regular four-sided figures, different either in form or colour".

Book of 500 Puzzles. 1859. Triangular problem, pp. 74-75. Identical to Hooper, dropping the last sentence.

Dudeney. CP. 1907. Prob. 74: The broken chessboard, pp. 119 121 & 220 221. The 12 pentominoes and a 2 x 2.

A. Aubry. Prob. 3224. Interméd. Math 14 (1907) 122-124. ??NYS -- cited by Grünbaum & Shephard who say Aubry has something of the idea or the term polyominoes.

G. Quijano. Prob. 3430. Interméd. Math 15 (1908) 195. ??NYS -- cited by Grünbaum & Shephard, who say he first asked for the number of n ominoes.

Thomas Scrutchin. US Patent 895,114 -- Puzzle. Applied: 20 Feb 1908; patented: 4 Aug 1908. 2pp + 1p diagrams. Mentioned in S&B, p. 18. A polyiamond puzzle -- triangle of side 8, hence with 64 triangles, apparently cut into 10 pieces (my copy is rather faint -- replace??).

Thomas W. Lancaster. US Patent 1,264,944 -- Puzzle. Filed: 7 May 1917; patented: 7 May 1918. 2pp + 1p diagrams. For a general polyrhomb puzzle making a rhombus. His diagram shows an 11 x 11 rhombus filled with 19 pieces formed from 4 to 10 rhombuses.

John Milner Lester. US Patent 1,290,761 -- Game Apparatus. Filed: 6 Feb 1918; patented: 7 Jan 1919. 2pp + 3pp diagrams. Fairly general assembly puzzle claims. He specifically illustrates a polyomino puzzle and a polyabolo puzzle. The first has a Greek cross of edge 3 (hence containing 45 unit cells) to be filled with polyominoes -- 11: 01154. The second has an 8-pointed star formed by superimposing two 4 x 4 squares. This has area 20 and hence contains 40 isosceles right triangles of edge 1, which is the basic unit of this type of puzzle. There are 11: 0128 pieces.

Blyth. Match-Stick Magic. 1921. Spots and squares, pp. 68-73. He uses matchsticks broken in thirds, so it is easier to describe with units of one-third. 6 units, 4 doubles and 2 triples. Some of the pieces have black bands or spots. Object is to form polyomino shapes without pieces crossing, but every intersection must have a black spot. 19 polyomino shapes are given to construct, including 7 of the pentominoes, though some of the shapes are only connected at corners.

"John Bull" Star of Fortune Prize Puzzle. 1922. This is a puzzle with 20 pieces, coloured red on one side, containing 6 through 13 triangles to be assembled into a star of David with 4 triangles along each edge (hence 12 x 16 = 192 triangles). Made by Chad Valley. Prize of £250 for a red star matching the key solution deposited at a bank; £150 for solution closest to the key; £100 for a solution with 10 red and 10 grey pieces, or as nearly as possible. Closing date of competition is 27 Dec 1922. Puzzle made by Chad Valley Co. as a promotional item for John Bull magazine, published by Odhams Press. A copy is in the toy shop of the Buckleys Shop Museum, Battle, East Sussex, to whom I am indebted for the chance to examine the puzzle and a photocopy of the puzzle, box and solution.

Daily Sketch Jig-Saw Puzzle. By Chad Valley. Card polyiamonds. 39: 0,0,1,5,6, 12,9,6, with a path printed on one side, to assemble into a shape of 16 rows of 15 with four corners removed and so the printed sides form a continuous circuit. In box with shaped bottom. Instructions on inside cover and loose sheet to submit solution. No dates given, but appears to be 1920s, though it is somewhat similar to the Daily Mail Crown Puzzle of 1953 -- cf below -- so it might be much later.

Daily Mirror Zig-Zag £500 Prize Puzzle. By Chad Valley. Card polyiamonds. 29: 0,0,1,1,4,  5,6,3,1,4, 1,2,1. One-sided pieces to fit into frame in card box. Three pieces are duplicated and one is triplicated. Solution and claim instructions appeared in Daily Mirror (17 Jan 1930) 1-2. See: Tom Tyler & Felicity Whiteley; Chad Valley Promotional Jig-Saw Puzzles; Magic Fairy Publishing, Petersfield, Hampshire, 1990, p. 55.

B. T.s Zig-Zag. B.T. is a Copenhagen newspaper. Polyiamond puzzle. 33: 0,0,1,2,5,  6,7,2,2,4,  1,1,1, Some repetitions, so I only see 20 different shapes. To be fit into an irregular frame. Solution given on 23 Nov 1931, pp. 1-2. (I have a photocopy of the form to fill in; an undated set of rules, apparently from the paper, saying the solutions must be received by 21 Nov; and the pages giving the solution; provided by Jan de Geus.)

Herbert D. Benjamin. Problem 1597: A big cutting-out design -- and a prize offer. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:9 (Dec 1934) 92. Finds the 35 hexominoes and asks if they form a 14 x 15 rectangle. Cites Dudeney (Tribune (20 Dec 1906)); Loyd (OPM (Apr-Jul 1908)) (see 6.F.1); Dudeney (CP, no. 74) (see above) and some other chessboard dissections. Jelliss says this is the first dissection problem in this journal.

F. Kadner. Solution 1597. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:10 (Feb 1935) 104-105. Shows the 35 hexominoes cannot tile a rectangle by two arguments, both essentially based on two colouring. Gives some other results and some problems are given as 1679-1681 -- ??NYS.

William E. Lester. Correction to 1597. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:11 (Apr 1935) 121. Corrects an error in Kadner. Finds a number of near-solutions. Editor says Kadner insists the editor should take credit for the two-colouring form of the previous proof.

Frans Hansson, proposer & solver?. Problem 1844. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:12 (Jun 1935) 128 & 2:13 (Aug 1935) 135. Finds both 3 x 20 pentomino rectangles.

W. E. Lester & B. Zastrow, proposers. Problem 1923. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:13 (Aug 1935) 138. Take an 8 x 8 board and remove its corners. Fill this with the 12 pentominoes.

H. D. Benjamin, proposer. Problem 1924. Problemist Fairy Chess Supplement (later called Fairy Chess Review) 2:13 (Aug 1935) 138. Dissect an 8 x 8 into the 12 pentominoes and the I-tetromino. Need solution -- ??NYS.

Thomas Rayner Dawson & William E. Lester. A notation for dissection problems. Fairy Chess Review 3:5 (Apr 1937) 46 47. Gives all n ominoes up to n = 6. Describes the row at a time notation. Shows the pentominoes and a 2 x 2 cover the chessboard with the 2 x 2 in any position. Asserts there are 108 7 ominoes and 368 8 ominoes -- citing F. Douglas & W. E. L[ester] for the hexominoes and J. Niemann for the heptominoes.

H. D. Benjamin, proposer. Problem 3228. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect a 5 x 5 into the five tetrominoes and a pentomino so that the pentomino touches all the tetrominoes along an edge. Asserts the solution is unique. Refers to problems 3026 3030 -- ??NYS.

H. D. Benjamin, proposer. Problem 3229. Fairy Chess Review 3:12 (Jun 1938) 129. Dissect an 8 x 8 into the 12 pentominoes and a tetromino so that all pieces touch the edge of the board. Asserts only one tetromino works.

T. R. D[awson], proposer. Problems 3230-1. Fairy Chess Review 3:12 (Jun 1938) 129. Extends prob. 3229 to ask for solutions with 12 pieces on the edge, using two other tetrominoes. Thinks it cannot be done with the remaining two tetrominoes.

Editorial note: The colossal count. Fairy Chess Review 3:12 (Jun 1938) 131. Describes progress on enumerating 8-ominoes (four people get 368 but Niemann gets 369) and 9 ominoes (numbers vary from 1237 to 1285). All workers are classifying them by the size of the smallest containing rectangle.

W. H. Rawlings, proposer. Problem 3930. Fairy Chess Review 4:3 (Nov 1939) 28. How many pentacubes are there? Ibid. 4:4 (Feb 1940) 75, reports that both 25 or 26 are claimed, but the editor has only seen 24. Ibid. 4:5 (Apr 1940) 85, reports that R. J. F[rench] has clearly shown there are 23 -- but this considers reflections as equal -- cf the 1948 editorial note.

R. J. French, proposer and solver. Problem 4149. Fairy Chess Review 4:3 (Nov 1939) 43 & 4:6 (Jun 1940) 93. Asks for arrangement of the pentominoes with the largest hole and gives one with 127 squares in the hole. (See: G. P. Jelliss; Comment on Problem 1277; JRM 22:1 (1990) 69. This reviews various earlier solutions and comments on Problem 1277.)

J. Niemann. Item 4154: "The colossal count". Fairy Chess Review 4:3 (Nov 1939) 44-45. Announces that there are 369 8-ominoes, 1285 9-ominoes and 4654 10-ominoes, but Keller and Jelliss note that he missed a 10 omino which was not corrected until 1966.

H. D. Benjamin. Unpublished notes. ??NYS -- cited and briefly described in G. P. Jelliss; Prob. 48 -- Aztec tetrasticks; G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss says Benjamin studied polysticks, which he called 'lattice dissections' around 1946-1948 and that some results by him and T. R. Dawson were entered in W. Stead's notebooks but nothing is known to have been published. For orders 1, 2, 3, 4, there are 1, 2, 5, 16 polysticks. Benjamin formed these into a 6 x 6 lattice square. Jelliss then mentions Barwell's rediscovery of them and goes on to a new problem -- see Knuth, 1999.

D. Nixon, proposer and solver. Problem 7560. Fairy Chess Review 6:16 (Feb 1948) 12 & 6:17 (Apr 1948) 131. Constructs 3 x 4 x 5 from solid pentominoes.

Editorial discussion: Space dissection. Fairy Chess Review 6:18 (Jun 1948) 141-142. Says that several people have verified the 23 pentacubes but that 6 of them have mirror images, making 29 if these are considered distinct. Says F. Hansson has found 77 6 cubes (these exclude mirror images and the 35 solid 6-ominoes). Gives many problems using n-cubes and/or solid polyominoes, which he calls flat n-cubes -- some are corrected in 7:2 (Oct 1948) 16 (erroneously printed as 108).

J. Niemann. The dissection count. Item 7803. Fairy Chess Review 7:1 (Aug 1948) 8 (erroneously printed as 100). Reports on counting n cubes. Gets the following.

n = 4 5 6 7

flat pieces 5 12 35 108

non-flats 2 11 77 499

TOTAL 7 23 112 607

mirror images 1 6 55 416

GRAND TOTAL 8 29 167 1023

R. J. French. Space dissections. Fairy Chess Review 7:2 (Oct 1948) 16 (erroneously printed as 108). French writes that he and A. W. Baillie have corrected the number of 6-cubes to 35 + 77 + 54 = 166. Baillie notes that every 6-cube lies in two layers -- i.e. has some width  2 -- and asks for the result for n cubes as prob. 7879. [I suspect the answer is that n  3k implies that an n-cube has some width  k.] Editor adds some corrections to the discussion in 6:18.

Editorial note. Fairy Chess Review 7:3 (Dec 1948) 23. Niemann and Hansson confirm the number 166 given in 7:2.

Daily Mail Crown Puzzle. Made by Chad Valley Co. 1953. 26 pieces, coloured on one side, to be fit into a crown shape. 11 are border pieces and easily placed. The other 15 are polyiamonds: 15: 00112 24012 11. Prize of £100 for solution plus best slogan, entries due on 8 Jun 1953.

S. W. Golomb. Checkerboards and polyominoes. AMM 61 (1954) 675 682. Mostly concerned with covering the 8 x 8 board with copies of polyominoes. Shows one covering with the 12 pentominoes and the square tetromino. Mentions that the idea can be extended to hexagons. S&B, p. 18, and Gardner (Dec 1964) say he mentions triangles, but he doesn't.

Walter S. Stead. Dissection. Fairy Chess Review 9:1 (Dec 1954) 2 4. Gives many pentomino and hexomino patterns -- e.g. one of each pattern of 8 x 8 with a 2 x 2 square deleted. "The possibilities of the 12 fives are not infinite but they will provide years of amusement." Includes 3 x 20, 4 x 15, 5 x 12 and 6 x 10 rectangles. No reference to Golomb. In 1955, Stead uses the 108 heptominoes to make a 28 x 28 square with a symmetric hole of size 28 in the centre -- first printed as cover of Chessics 28 (1986).

Jules Pestieau. US Patent 2,900,190 -- Scientific Puzzle. Filed: 2 Jul 1956; patented: 18 Aug 1959. 2pp + 1p diagrams. For the 12 pentominoes! Diagram shows the 6 x 10 solution with two 5 x 6 rectangles and shows the two-piece non-symmetric equivalence of the N and F pieces. Pieces have markings on one side which may be used -- i.e. pieces may not be turned over. Mentions possibility of using n-ominoes.

Gardner. SA (Dec 1957) = 1st Book, chap. 13. Exposits Golomb and Stead. Gives number of n-ominoes for n = 1, ..., 7. 1st Book describes Scott's work. Says a pentomino set called 'Hexed' was marketed in 1957. (John Brillhart gave me and my housemates an example in 1960 -- it took us two weeks to find our first solution.)

Dana Scott. Programming a Combinatorial Puzzle. Technical Report No. 1, Dept. of Elec. Eng., Princeton Univ., 1958, 20pp. Uses MANIAC to find 65 solutions for pentominoes on an 8 x 8 board with square 2 x 2 in the centre. Notes that the 3 x 20 pentomino rectangle has just two solutions. In 1999, Knuth notes that the total number of solutions with the 2 x 2 being anywhere does not seem to have ever been published and he finds 16146.

M. Gardner. SA (Sep 1958) c= 2nd Book, chap. 6. First general mention of solid pentominoes, pentacubes, tetracubes. In the Addendum in 2nd Book, he says Theodore Katsanis of Seattle suggested the eight tetracubes and the 29 pentacubes in a letter to Gardner on 23 Sep 1957. He also says that Julia Robinson and Charles W. Stephenson both suggested the solid pentominoes.

C. Dudley Langford. Note 2793: A conundrum for form VI. MG 42 (No. 342) (Dec 1958) 287. 4 each of the L, N, and T (= Y) tetrominoes make a 7 x 7 square with the centre missing. Also nine pieces make a 6 x 6 square but this requires an even number of Ts.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 8. Use the pentominoes to make two 5 x 5 squares at the same time. Solution just says there are several ways to do so.

J. C. P. Miller. Pentominoes. Eureka 23 (Oct 1960) 13 16. Gives the Haselgroves' number of 2339 solutions for the 6 x 10 and says there are 2 solutions for the 3 x 20. Says Lehmer suggests assembling 12 solid pentominoes into a 3 x 4 x 5 and van der Poel suggests assembling the 12 hexiamonds into a rhombus.

C. B. & Jenifer Haselgrove. A computer program for pentominoes. Ibid., 16 18. Outlines program which found the 2339 solutions for the 6 x 10. It is usually said that they also found all solutions of the 3 x 20, 4 x 15 and 5 x 12, but I don't see it mentioned here and in JRM 7:3 (1974) 257, it is reported that Jenifer (Haselgrove) Leech stated that only the 6 x 10 and 3 x 20 were done in 1960, but that she did the 5 x 12 and 4 x 15 with a new program in c1966. See Fairbairn, c1962, and Meeus, 1973.

Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.] 7 (Oct & Dec 1960) 97-106 & 144-153. Considers handed polyominoes, i.e. polyominoes when reflections are not considered equivalent. Notes that neither the 5 plain nor the 7 handed tetrominoes can form a rectangle. The 10 chequered handed tetrominoes form 4 x 10 and 5 x 8 rectangles and he has several solutions of each. There is no 2 x 20 rectangle. Discusses MacMahon pieces -- cf 5.H.2 -- and polyiamonds. He uses the word 'hexiamond', but not 'polyiamond' -- in an email of 8 Apt 2000, Guy says that O'Beirne invented all the terms. He considers making a 'hexagon' from the 19 hexiamonds. Part II considers solid problems and uses the term 'solid pentominoes'.

Solomon W. Golomb. The general theory of polyominoes: part 2 -- Patterns and polyominoes. RMM 5 (Oct 1961) 3-14. ??NYR.

J. E. Reeve & J. A. Tyrrell. Maestro puzzles. MG 45 (No. 353) (Oct 1961) 97 99. Discusses hexiamond puzzles, using the 12 reversible pieces. [The puzzle was marketed under the name 'Maestro' in the UK.]

T. H. O'Beirne. Pell's equation in two popular problems. New Scientist 12 (No. 258) (26 Oct 1961) 260 261.

T. H. O'Beirne. Pentominoes and hexiamonds. New Scientist 12 (No. 259) (2 Nov 1961) 316 317. This is the first use of the word 'polyiamond'. He considers the 19 one sided pieces. He says he devised the pieces and R. K. Guy has already published many solutions in Nabla. He asks for the number of ways the 18 one-sided pentominoes can fill a 9 x 10. In 1999, Knuth found this would take several months.

T. H. O'Beirne. Some hexiamond solutions: and an introduction to a set of 25 remarkable points. New Scientist 12 (No. 260) (9 Nov 1961) 378 379.

Maurice J. Povah. Letter. MG 45 (No. 354) (Dec 1961) 342. States Scott's result of 65 and the Haselgroves' result of 2339 (computed at Manchester). Says he has over 7000 solutions for the 8 x 8 board using a 2 x 2.

T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751 752.

T. H. O'Beirne. Some tetrabolic difficulties. New Scientist 13 (No. 270) (18 Jan 1962) 158 159. These two columns are the first mention of tetraboloes, so named by S. J. Collins.

R. A. Fairbairn. Pentomino Problems: The 6 x 10, 5 x 12, 4 x 15, and 3 x 20 Rectangles -- The Complete Drawings. Unpublished MS, undated, but c1962, based on the Haselgroves' work of 1960. ??NYS -- cited by various authors, e.g. Madachy (1969), Torbijn (1969), Meeus (1973). Madachy says Fairbairn is from Willowdale, Ontario, and takes some examples from his drawings. However, the dating is at variance with Jenifer Haselgrove's 1973 statement - cf Haselgrove, 1960. Perhaps this MS is somewhat later?? Does anyone know where this MS is now? Cf Meeus, 1973.

Serena Sutton Besley. US Patent 3,065,970 -- Three Dimensional Puzzle. Filed: 6 Jul 1960; issued: 27 Nov 1962. 2pp + 4pp diagrams. For the 29 pentacubes, with one piece duplicated giving a set of 30. Klarner had already considered omitting the 1 x 1 x 5 and found that he could make two separate 2 x 5 x 7s. Besley says the following can be made: 5 x 5 x 6, 3 x 5 x 10, 2 x 5 x 15, 2 x 3 x 25; 3 x 5 x 6, 3 x 3 x 10, 2 x 5 x 9, 2 x 3 x 15; 3 x 4 x 5, 2 x 5 x 6, 2 x 3 x 10 (where the latter three are made with the 12 solid pentominoes and the previous four are made with the 18 non-planar pentacubes) but detailed solutions are only given for the 5 x 5 x 6, 3 x 5 x 6, 3 x 4 x 5. Mentions possibility of n-cubes.

M. Gardner. Polyiamonds. SA (Dec 1964) = 6th Book, chap. 18. Exposits basic ideas and results for the 12 double sided hexiamonds. Poses several problems which are answered by readers. The six-pointed star using 8 pieces has a unique solution. John G. Fletcher and Jenifer (Haselgrove) Leech both showed the 3 x 12 rhombus is impossible. Fletcher found the 3 x 11 rhombus has 24 solutions, all omitting the 'bat'. Leech found 155 solutions for the 6 x 6 rhombus and 74 solutions for the 4 x 9. Mentions there are 160 9-iamonds, one with a hole.

John G. Fletcher. A program to solve the pentomino problem by the recursive use of macros. Comm. ACM 8 (1965) 621-623. ??NYS -- described by Knuth in 1999 who says that Fletcher found the 2339 solutions for the 6 x 10 in 10 minutes on an IBM 7094 and that the program remains the fastest known method for problems of placing the 12 pentominoes.

M. Gardner. Op art. SA (Jul 1965) = 6th Book, chap. 24. Shows the 24 heptiamonds and discusses which will tile the plane.

Solomon W. Golomb. Tiling with polyominoes. J. Combinatorial Theory 1 (1966) 280-296. ??NYS. Extended by his 1970 paper.

T. R. Parkin. 1966. ??NYS -- cited by Keller. Finds 4655 10-ominoes.

M. Gardner. SA (Jun 1967) = Magic Show, chap. 11. First mention of polyhexes.

C. J. Bouwkamp. Catalogue of Solutions of the Rectangular 3 x 4 x 5 Solid Pentomino Problem. Dept. of Math., Technische Hogeschool Eindhoven, July 1967, reprinted 1981, 310pp.

C. J. Bouwkamp. Packing a rectangular box with the twelve solid pentominoes. J. Combinatorial Thy. 7 (1969) 278 280. He gives the numbers of solutions for rectangles as 'known'.

2 x 3 x 10 can be packed in 12 ways, which are given.

2 x 5 x 6 can be packed in 264 ways.

3 x 4 x 5 can be packed in 3940 ways. (See his 1967 report.)

T. R. Parkin, L. J. Lander & D. R. Parkin. Polyomino enumeration results. Paper presented at the SIAM Fall Meeting, Santa Barbara, 1 Dec 1967. ??NYS -- described by Madachy, 1969. Gives numbers of n-ominoes, with and without holes, up to n = 15, done two independent ways.

Joseph S. Madachy. Pentominoes -- Some solved and unsolved problems. JRM 2:3 (Jul 1969) 181-188. Gives the numbers of Parkin, Lander & Parkin. Shows various examples where a rectangle splits into two congruent halves. Discusses various other problems, including Bouwkamp's 3 x 4 x 5 solid pentomino problem. Bouwkamp reports that the final total of 3940 was completed on 16 Mar 1967 after about three years work using three different computers, but that a colleague's program would now do the whole search in about three hours.

P. J. Torbijn. Polyiamonds. JRM 2:4 (Oct 1969) 216-227. Uses the double sided hexiamonds and heptiamonds. A few years before, he found, by hand, that there are 156 ways to cover the 6 x 6 rhombus with the 12 hexiamonds and 74 ways for the 4 x 9, but could find no way to cover the 3 x 12. The previous year, John G. Fletcher confirmed these results with a computer and he displays all of these -- but this contradicts Gardner (Dec 64) -- ?? He gives several other problems and results, including using the 24 heptiamonds to form 7 x 12, 6 x 14, 4 x 21 and 3 x 28 rhombuses.

Solomon W. Golomb. Tlling with sets of polyominoes. J. Combinatorial Theory 9 (1970) 60 71. ??NYS. Extends his 1966 paper. Asks which heptominoes tile rectangles and says there are two undecided cases -- cf Marlow, 1985. Gardner (Aug 75) says Golomb shows that the problem of determining whether a given finite set of polyominoes will tile the plane is undecidable.

C. J. Boukamp & D. A. Klarner. Packing a box with Y-pentacubes. JRM 3:1 (1970) 10-26. Substantial discussion of packings with Y pentominoes and Y-pentacubes. Smallest boxes are 5 x 10 and 2 x 5 x 6 and 3 x 4 x 5.

Fred Lunnon. Counting polyominoes. IN: Computers in Number Theory, ed. by A. O. L. Atkin & B. J. Birch; Academic Press, 1971, pp. 347-372. He gets up through 18 ominoes, but the larger ones can have included holes. The numbers for n = 1, 2, ..., are as follows: 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107911, 192622052. These values have been quoted numerous times.

Fred Lunnon. Counting hexagonal and triangular polyominoes. IN: Graph Theory and Computing, ed. by R. C. Read; Academic Press, 1972, pp. 87-100. ??NYS -- cited by Grünbaum & Shephard.

M. Gardner. SA (Sep 1972). c= Knotted, chap. 3. Says the 8 tetracubes were made by E. S. Lowe Co. in Hong Kong and marketed as "Wit's End". Says an MIT group found 1390 solutions for the 2 x 4 x 4 box packed with tetracubes. He reports that several people found that there are 1023 heptacubes -- but see Niemann, 1948, above. Klarner reports that the heptacubes fill a 2 x 6 x 83.

Jean Meeus. Some polyomino and polyamond [sic] problems. JRM 6:3 (1973) 215-220. (Corrections in 7:3 (1974) 257.) Considers ways to pack a 5 x n rectangle with some n pentominoes. A. Mank found the number of ways for n = 2, 3, ..., 11 as follows, and the number for n = 12 was already known:

0,   7,  50, 107, 541, 1387, 3377, 5865,   6814,   4103, 1010.

Says he drew out all the solutions for the area 60 rectangles in 1972 (cf Fairbairn, c1962). Finds that 520 of the 6 x 10 rectangles can be divided into two congruent halves, sometimes in two different ways. For 5 x 12, there are 380; for 4 x 15, there are 94. Gives some hexomino rectangles by either deleting a piece or duplicating one, and an 'almost 11 x 19'. Says there are 46 solutions to the 3 x 30 with the 18 one-sided pentominoes and attributes this to Mrs (Haselgrove) Leech, but the correction indicates this was found by A. Mank.

Jenifer Haselgrove. Packing a square with Y-pentominoes. JRM 7:3 (1974) 229. She finds and shows a way to pack 45 Y-pentominoes into a 15 x 15, but is unsure if there are more solutions. In 1999, Knuth found 212 solutions. She also reports the impossibility of using the Y-pentominoes to fill various other rectangles.

S. W. Golomb. Trademark for 'PENTOMINOES'. US trademark 1,008,964 issued 15 Apr 1975; published 21 Jan 1975 as SN 435,448. (First use: November 1953.) [These appear in the Official Gazette of the United States Patent Office (later Patent and Trademark Office) in the Trademarks section.]

M. Gardner. Tiling with polyominoes, polyiamonds and polyhexes. SA (Aug 75) (with slightly different title) = Time Travel, chap. 14. Gives a tiling criterion of Conway. Describes Golomb's 1966 & 1970 results.

C. J. Bouwkamp. Catalogue of solutions of the rectangular 2 x 5 x 6 solid pentomino problem. Proc. Koninklijke Nederlandse Akad. van Wetenschappen A81:2 (1978) 177 186. Presents the 264 solutions which were first found in Sep 1967.

H. Redelmeier. Discrete Math. 36 (1981) 191 203. ??NYS -- described by Jelliss. Obtains number of n ominoes for n  24.

Karl Scherer. Problem 1045: Heptomino tessellations. JRM 14:1 (1981 82) 64. XX 

Says he has found that the heptomino at the right fills a 26 x 42 rectangle. XXXXX

See Dahlke below.

David Ellard. Poly-iamond enumeration. MG 66 (No. 438) (Dec 1982) 310 314. For n = 1, ..., 12, he gets 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3342 n-iamonds. One of the 8-iamonds has a hole and there are many later cases with holes.

Anon. 31: Polyominoes. QARCH 1:8 (June 1984) 11 13. [This is an occasional publication of The Archimedeans, the student maths society at Cambridge.] Good survey of counting and asymptotics for the numbers of polyominoes, up to n = 24, polycubes, etc. 10 references.

T. W. Marlow. Grid dissections. Chessics 23 (Autumn, 1985) 78 79.

X XX

Shows XXXXX fills a 23 x 24 and XXXXX fills a 19 x 28.

Herman J. J. te Riele & D. T. Winter. The tetrahexes puzzle. CWI Newsletter [Amsterdam] 10 (Mar 1986) 33 39. Says there are: 7 tetrahexes, 22 pentahexes, 82 hexahexes, 333  heptahexes, 1448 octahexes. Studies patterns of 28 hexagons. Shows the triangle cannot be constructed from the 7 tetrahexes and gives 48 symmetric patterns that can be made.

Karl A. Dahlke. Science News 132:20 (14 Nov 1987) 310. (??NYS -- cited in JRM 21:3 and

XX

22:1 and by Marlow below.) Shows XXXXX fills a 21 x 26 rectangle.

The results of Scherer and Dahlke are printed in JRM 21:3 (1989) 221 223 and Dahlke's solution is given by Marlow below.

Karl A. Dahlke. J. Combinatorial Theory A51 (1989) 127 128. ??NYS -- cited in JRM 22:1. Announces a 19 x 28 solution for the above heptomino problem, but the earlier 21 x 26 solution is printed by error. The 19 x 28 solution is printed in JRM 22:1 (1990) 68 69.

Tom Marlow. Grid dissections. G&PJ 12 (Sep/Dec 1989) 185. Prints Dahlke's result.

Brian R. Barwell. Polysticks. JRM 22:3 (1990) 165-175. Polysticks are formed of unit lengths on the square lattice. There are: 1, 2, 5, 16, 55 polysticks formed with 1, 2, 3, 4, 5 unit lengths. He forms 5 x 5 squares with one 4-stick omitted, but he permits pieces to cross. He doesn't consider the triangular or hexagonal cases. See also Blyth, 1921, for a related puzzle. Cf Benjamin, above, and Wiezorke & Haubrich, below.

General Symmetrics (Douglas Engel) produced a version of polysticks, ©1991, with 4 3 sticks and 3 4-sticks to make a 3 x 3 square array with no crossing of pieces.

Nob Yoshigahara. Puzzlart. Tokyo, 1992. Ip-pineapple (pineapple delight), pp. 78-81. Imagine a cylindrical solution of the 6 x 10 pentomino rectangle and wrap it around a cylinder, giving each cell a depth of one along the radius. Hence each cell is part of an annulus. He reduces the dimensions along the short side to make the cells look like tenths of a slice of pineapple. Nob constructed and example for Toyo Glass's puzzle series and it was later found to have a unique solution.

Kate Jones, proposer; P. J. Torbijn, Jacques Haubrich, solvers. Problem 1961 -- Rhombiominoes. JRM 24:2 (1992) 144-146 & 25:3 (1993) 223 225. A rhombiomino or polyrhomb is a polyomino formed using rhombi instead of squares. There are 20 pentarhombs. Fit them into a 10 x 10 rhombus. Various other questions. Haubrich found many solutions. See Lancaster, 1918.

Bernard Wiezorke & Jacques Haubrich. Dr. Dragon's polycons. CFF 33 (Feb 1994) 6-7. Polycons (for connections) are the same as the polysticks described by Barwell in 1990, above. Authors describe a Taiwanese version on sale in late 1993, using 10 of the 4 sticks suitably shortened so they fit into the grooves of a 4 x 4 board -- so crossings are not permitted. (An n x n board has n+1 lines of n edges in each direction.) They fit 15 of the 4-sticks onto a 5 x 5 board and determine all solutions.

CFF 35 (Dec 1994) 4 gives a number of responses to the article. Brain Barwell wrote that he devised them as a student at Oxford, c1970, but did not publish until 1990. He expected someone to say it had been done before, but no one has done so. He also considered using the triangular and hexagonal lattice. He had just completed a program to consider fitting 15 of the 4-sticks onto a 5 x 5 board and found over 180,000 solutions, with slightly under half having no crossings, confirming the results of Wiezorke & Haubrich.

Dario Uri also wrote that he had invented the idea in 1984 and called them polilati (polyedges). Giovanni Ravesi wrote about them in Contromossa (Nov 1984) 23 -- a defunct magazine.

Chris Roothart. Polylambdas. CFF 34 (Oct 1994) 26-28. A lambda is a 30^o-60^o-90^o triangle. These may be joined along corresponding legs, but not along hypotenuses. For n = 1, 2, 3, 4, 5, there are 1, 4, 4, 11, 12 n-lambdas. He gives some problems using various sets of these pieces.

Richard Guy. Letters of 29 May and 13 Jun 1996. He is interested in using the 19 one-sided hexiamonds. Hexagonal rings of hexagons contain 1, 6, 12 hexagons, so the hexagon with three hexagons on a side has 19 hexagons. If these hexagons are considered to comprise six equilateral triangles, we have a board with 19 x 6 triangles. O'Beirne asked for the number of ways to fill this board with the one-sided hexiamonds. Guy has collected over 4200 solutions. A program by Marc Paulhus found 907 solutions in eight hours, from which it initially estimated that there are about 30,000 solutions. The second letter gives the final results -- there are 124,518 solutions. This is modulo the 12 symmetries of the hexagon. In 1999, Knuth found 124,519 and Paulhus has rerun his program and found this number.

Ferdinand Lammertink. Polyshapes. Parts 1 and 2. The author, Hengelo, Netherlands, 1996 & 1997. Part 1 deals with two dimensional puzzles. Good survey of the standard polyform shapes and many others.

Hilarie Korman. Pentominoes: A first player win. IN: Games of No Chance; ed. by Richard Nowakowski; CUP, 1997??, ??NYS - described in William Hartston; What mathematicians get up to; The Independent Long Weekend (29 Mar 1997) 2. This studies the game proposed by Golomb -- players alternately place one of the pentominoes on the chess board, aligned with the squares and not overlapping the previous pieces, with the last one able to play being the winner. She used a Sun IPC Sparcstation for five days, examining about 22 x 10⁹ positions to show the game is a first player win.

Nob Yoshigahara found in 1994 that the smallest box which can be packed with W-pentacubes is 5 x 6 x 6. In 1997, Yoshya (Wolf) Shindo found that one can pack the 6 x 10 x 10 with Z-pentacubes, but it is not known if this is the smallest such box. These were the last unsolved problems as to whether a box could be packed with a planar pentacube (= solid pentomino).

Marcel Gillen & Georges Philippe. Twinform 462 Puzzles in one. Solutions for Gillen's puzzle exchange at IPP17, 1997, 32pp + covers. Take 6 of the pentominoes and place them in a 7 x 5 rectangle, then place the other six to make the same shape on top of the first shape. There are 462 (= BC(12,6)/2) possible puzzles and all of them have solutions. Taking F, T, U, W, X, Z for the first layer, there is just one solution; all other cases have multiple solutions, totalling 22,873 solutions, but only one solution for each case is given here.)

Richard K. Guy. O'Beirne's hexiamond. In: The Mathemagican and Pied Puzzler; ed. by Elwyn Berlekamp & Tom Rodgers, A. K. Peters, Natick, Massachusetts, 1999, pp. 85 96. He relates that O'Beirne discovered the 19 one-sided hexiamonds in c1959 and found they would fill a hexagonal shape in Nov 1959 and in Jan 1960 he found a solution with the hexagonal piece in the centre. He gives Paulhus's results (see Guy's letters of 1996), broken down in various ways. He gives the number of double-sided (i.e. one can turn them over) and single-sided n-iamonds for n = 1, ..., 7. Cf Ellard, 1982, for many more values for the double-sided case.

n 1 2 3 4 5 6 7

double 1 1 1 3 4 12 24

single 1 1 1 4 6 19 44

In 1963, Conway and Mike Guy considered looking for 'symmetric' solutions for filling the hexagonal shape with the 19 one-sided hexiamonds. A number of these are described.

Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by the author. Available as: http://www-cs-faculty.stanford.edu/~knuth/preprints.html . In this he introduces a new technique for backtrack programming which runs faster (although it takes more storage) and is fairly easy to adapt to different problems. In this approach, there is a symmetry between pieces and cells. He applies it to several polyshape problems, obtaining new, or at least unknown, results. He extends Scott's 1958 results to get 16146 ways to pack the 8 x 8 with the 12 pentominoes and the 2 x 2. He describes Fletcher's 1965 work. He extends Haselgrove's 1974 work and finds 212 ways to fit 15 Y-pentominoes in a 15 x 15. Describes Torbijn's 1969 work and Paulhus' 1996 work on hexiamonds, correcting the latter's number to 124,519. He then looks for the most symmetric solutions for filling the hexagonal shape with the 19 one-sided hexiamonds, in the sense discussed by Guy (1999). He then considers the 18 one-sided pentominoes (cf Meeus (1973)) and tries the 9 x 10, but finds it would take a few months on his computer (a 500 MHz Pentium III), so he's abandoned it for now. He then considers polysticks, citing an actual puzzle version that I've not seen. He adapts his program to them. He considers the 'welded tetrasticks' which have internal junction points. There are six of these and ten if they are taken as one-sided. The ten can be placed in a 4 x 4 grid. There are 15 unwelded, one-sided, tetrasticks, but they do not form a square, nor indeed any nice shape. He considers all 25 one-sided tetrasticks and asks if they can be fit into what he calls an Aztec Diamond, which is the shape looking like a square tilted 45^o on the square lattice. The rows contain 1, 3, 5, 7, 9, 7, 5, 3, 1 cells. He thinks an exhaustive search is beyond present computing power.

G. P. Jelliss. Prob. 48 -- Aztec tetrasticks. G&PJ 2 (No. 17) (Oct 1999) 320. Jelliss first discusses Benjamin's work on polysticks (see at 1946-1948 above) and Barwell's rediscovery of them (see above). He then describes Knuth's Dancing Links and gives the Aztec Diamond problem. Jelliss has managed to get all but one of the polysticks into the shape, but feels it is impossible to get them all in.

Harry L. Nelson. Solid pentomino storage, Question and answer. 1p HO at G4G5, 2002. 1: Can one put all the solid pentominoes into a cube of edge 4.5? What is the smallest cube into which they can all be placed? He gives 2 solutions to 1 and a solution due to Wei-Hwa Huang for a cube of edge 4.405889..., which is conjectured to be minimal. In fact, one edge of the packing is actually 4, so the volume is less than (4.405889...)³. This leads me to ask what is the smallest volume of a cuboid, with edges less than 5, that contains all the solid pentominoes. In Summer 2002, Harry gave me a set of solid pentominoes in a box with a list of various rectangles and boxes to fit them into: 3 x 22; 3 x 21; 3 x 20; 4 x 16; 4 x 15; 5 x 13; 5 x 12; 6 x 11; 6 x 10; 7 x 9; 8 x 8; 2 x 4 x 8; 2 x 5 x 7; 2 x 5 x 6; 2 x 6 x 6; 3 x 4 x 6; 3 x 4 x 5; 3 x 5 x 5; 4 x 4 x 5; the given box: 4.4 x 4.4 x 4.9.)