7. arithmetic & number theoretic recreations a. Fibonacci numbers

No. 9: The "sixty-five" puzzle. 5 x 5 case.

Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S. National Museum for 1893, pp. 489 537. On pp. 536-537, he discusses the Lok Shü, citing Legge. He reports that the 618 version is popular as a charm with both Hindus and Moslems in India, while the Chinese 492 form is used in Tibet.

Dudeney. A batch of puzzles. Royal Magazine 1:3 (Jan 1899) & 1:4 (Feb 1899) 368-372. The eight clowns puzzle. = CP; 1907; prob. 81, pp. 128 & 126. 3 x 3 array with pieces x 2 3; 4 5 6; 7 8 9 to be rearranged into a magic square, the blank being counted as a 0. Answer is that this is impossible, but the clown marked 9 is juggling balls which make his number .9, i.e. .9 recurring, which is 1!

Dudeney. The magic square of sixteen. The Queen (15 Jan 1910) 125-126. Good derivation of the 880 squares of order 4, classified into 12 types. A condensed version with some extra information is in AM, pp. 119-121.

Loyd. Cyclopedia. 1914. The 14 15 puzzle in puzzleland, pp. 235 & 371. = MPSL1, prob. 21, pp. 19 20 & 128. c= SLAHP: The "14 15" magic square, pp. 17 18 & 89. Given the 15 Puzzle with the 14 and 15 interchanged, move to a magic square. The blank counts zero, so the magic constant is 30.

Collins. Book of Puzzles. 1927. Magic squares and other figures, pp. 79-94. Brief survey. Gives a number of variant forms.

D. N. Lehmer. A complete census of 4 x 4 magic squares. Bull. Amer. Math. Soc. 39 (1933) 764-767. Here he is dealing with semi-magic squares and then any permutation of the rows or columns or transposition of the whole array preserves the row and column sums. Hence there are 2(n!)² arrays in each equivalence class and he describes a normalized form for each class. For the 3 x 3 case, there are 72 semimagic squares and one normalized form. For the 4 x 4 case, there are 468 normalized forms and hence 468 x 2 x 24² = 539,136 semimagic squares.

D. N. Lehmer. A census of squares of order 4, magic in rows, columns, and diagonals. Bull. Amer. Math. Soc. 39 (1933) 981-982. Here he discusses Frenicle's enumeration of 880 4 x 4 magic squares and points out that there are additional equivalences beyond the symmetries of the square so that Frenicle only needed to find and list 220 squares. Using ideas like those in his previous article, he finds 220 such squares, confirming Frenicle's result once again.

J. Travers. Rules for bordered magic squares. MG 23 (No. 256) (Oct 1939) 349 351. Cites Rouse Ball, MRE (no details), as saying no such rules are known. He believes these are the first published rules.

Anonymous. A Book of Fun with Games and Puzzles. One of a set of three 12pp booklets, no details, [1940s?]. P. 7: Here is the magic square. Gives a 4 x 4 square using the numbers 3, ..., 17 and asks for it to be dissected along the lines into four pieces which can be rearranged into a magic square.

D. H. Hallowes. On 4 x 4 pan magic squares. MG 30 (No. 290) (Jul 1946) 153 154. Shows there are only 3 essentially different 4 x 4 pan magic squares.

Ripley's Puzzles and Games. 1966. P. 52. The magic cube. This actually shows only the front of a cube and really comprises three magic squares as the faces have no relation to each other. What is really being attempted is to arrange the numbers 1, 2, ..., 27 into three 3 x 3 magic squares. Then the magic sum must be 42. The given arrangement has 22 of the 24 lines adding to 42 -- two of the diagonals fail. ?? -- is such an arrangement possible? Ripley's says the pattern adds to 42 in 44 directions -- apparently they count each of the 22 lines in each direction.

Gardner. SA (Jan 1976) c= Time Travel, chap. 17. Richard Schroeppel, of Information International, used a PDP 10 to find 275,305,224 magic squares of order 5, inequivalent under the 8 symmetries of the square. If one also considers the 'eversion' symmetries, there are 32 symmetries and 68,826,306 inequivalent squares. (Gardner says there is an Oct 1975 report on this work by Michael Beeler, ??NYS, and gives Schroeppel's address: 835 Ashland Ave., Santa Monica, Calif., 90405 -- I believe I wrote, but had no reply??)

K. Ollerenshaw & H. Bondi. Magic squares of order four. Phil. Trans. Roy. Soc. Lond. A306 (1982) 443 532. (Also available separately.) Gives a new approach to Frénicle's results. Relates to Magic Card Squares and the Fifteen Puzzle.

Lee C. F. Sallows. Alphamagic squares: I & II. Abacus 4:1 (Fall 1986) 28 45 & 4:2 (Winter 1987) 20-29 & 43. Reprinted in: The Lighter Side of Mathematics; ed. by R. K. Guy & R. E. Woodrow; MAA, 1994, pp. 305-339. Introduces notion of alphamagic square -- a magic square such that the numbers of letters in the words for the numbers also form a magic square. Simplest example is: 5, 22, 18; 28, 15, 2; 12, 8, 25. He asserts that this appears in runes in an 1888 book describing a 5C charm revealed to King Mi of North Britain. (This seems a bit far fetched to me or mi?) Asks if there can be an alphamagic square using the first n² numbers and shows that n  14. Notes some interesting results on formulae for 4 x 4 squares, including one with minimum number of symbols. There was a letter and response in Abacus 4:3 (Spring 1987) 67-69.

Martin Gardner. Prime magic squares. IN: The Mathematical Sciences Calendar for 1988; ed. by Nicholas J. Rose, Rome Press, Raleigh, North Carolina, 1987. Reprinted with postscript in Workout, chap. 25. Says Akio Suzuki found a 35 x 35 magic square using the first odd primes in 1957. I have a poster of this which Gardner gave me. Offers $100 for the first 3 x 3 magic square using consecutive primes. The postscript says Harry L. Nelson won, using a Cray at Lawrence Livermore Laboratories. He found 22 examples. The one with the lowest constant has smallest value 1,480,028,129 and the values all have the same first seven digits and their last three digits are: 129,  141,  153,  159, 171, 183, 189, 201, 213.

Lalbhai D. Patel. The secret of Franklin's 8 x 8 'magic' square. JRM 23:3 (1991) 175-182. Develops a method to make Franklin's squares as fast as one can write down the numbers!

Jacques Bouteloup. Carrés Magiques, Carrés Latins et Eulériens. Éditions du Choix, Bréançon, 1991. Nice systematic survey of this field, analysing many classic methods.

Lee Sallows. Alphamagic squares. CFF 35 (Dec 1994) 6-10. "... a brief synopsis of [his above article] which handles the topic in very much greater detail."

Martin Gardner. The magic of 3 x 3. Quantum 6:3 (Jan-Feb 1996) 24-26; with addendum in (Mar-Apr 1996). Reprinted with a postscript in Workout, chap. 22. Says that modern scholars doubt if the pattern in China is older than 10C! Mentions his 1987 prize and Nelson's least solution. Says Martin LaBar [CMJ (Jan 1984) 69] asked for a 3 x 3 magic square whose entries are all positive squares. Gardner reiterates this and offers $100 for the first example -- in the postscript, he extends the prize to include a proof of impossibility. He gives examples of 3 x 3 squares with various properties and the lowest magic sum, e.g. using primes in arithmetic progression.

Lee Sallows. The lost theorem. Math. Intell. 19:4 (1997) 51-54. Gives an almost solution to Gardener's problem, but one diagonal fails to add up correctly. Gives an example of Michael Schweitzer which is magic but contains only six squares. Using Lucas' pattern, where the central number, c, is one third of the magic sum and two adjacent corners are c + a and c + b, he observes that these can be vectors in the plane or complex numbers, which allows one to correspond classes of eight magic squares with parallelograms in the plane. This leads to perhaps the most elegant magic square by taking c = 0, a = 1, b = i.

Kathleen Ollerenshaw, Kathleen & David S. Brée. Most-perfect Pandiagonal Magic Squares Their construction and enumeration. Institute of Mathematics and its Applications, Southend-on-Sea, 1998. A most-perfect square is one which is magic and pandiagoanal and all 2 x 2 subsquares have the same sum, even when the square is considered on a torus. They find a formula for the number of these in general. Summary available on www.magic-squares.com or www.most-perfect.com .

Lee Sallows. Email of 11 Jun 1998. He asks if any set of 16 distinct numbers can produce more than 880 magic squares of order 4. He finds that -8, -7, ..., -2, -1, 1, 2, ..., 7, 8 gives 1040 magic squares. He has not found any other examples, nor indeed any new examples with as many as 880, though he has looked at other types of values, even Gaussian integers.

K. Pinn & C. Wieczerkowski. Number of magic squares from parallel tempering Monte Carlo. Intern. J. Modern Physics C 9:4 (1998) 541-546. ??NYS - cited by Chan & Loly, below. They estimate there are 1.77 x 10¹⁹ magic squares of order 6.

Harvey D. Heinz & John R. Hendricks. Magic Square Lexicon: Illustrated. Harvey D. Heinz, 15450 92A Avenue, Surrey, British Columbia, 2000.

Frank J. Swetz. Legacy of the Luoshu - The 4000 Year Search for the Meaning of the Magic Square of Order Three. Open Court, 2002. ??NYS - cited by Chan & Loly, below.

Wayne Chan & Peter Loly. Iterative compounding of square matrices to generate large-order magic squares. Mathematics Today 38:4 (Aug 2002) 113-118. Primarily they develop programs for doing compounding to produce very large squares.

Lee Sallows. Christmas card for 2003: Geometric magic square. Consider the magic square:

11 9 9 7; 6 10 8 12; 6 8 10 12; 13 9 9 5 with magic constant 36. Each quadrant also adds up to 36. Sallows uses sixteen polyominoes having these numbers of unit squares and arranged in this pattern so that each quadrant forms a 6 x 6 square. These polyominoes can be assembled into many other squares.

7.N.1 MAGIC CUBES
Note. Historically, a k³ has been called magic when all the 3k² lines parallel to the axes and the 4 space diagonals have the same sum. But there are also 6k 2 dimensional diagonals -- if these also have the same sum, we will say that the cube is perfectly magic. Pandiagonal (= pan n agonal) refers just to the space diagonals. Perfectly pandiagonal refers to all the diagonals. In higher space, the simpler words refer to the 2^n-1 'space' diagonals and perfect will include all the diagonals in intermediate dimensions.

A k-agonal is a line which varies in k coordinates, so a 1-agonal is a row or column, etc., the 2-agonals of a cube include the diagonals of the faces, while the 3-agonals of a 3 cube are the space diagonals.

Associated or complementary means that two cells symmetric with respect to the centre add to kⁿ + 1.
Pierre de Fermat. Letter to Mersenne (1 Apr 1640). Oeuvres de Fermat. Ed. by P. Tannery & C. Henry. Vol. 2, Gauthier Villars, Paris, 1894, pp. 186 194. Gives a magic 4³.

A shorter, undated, version, occurs in Varia Opera Mathematica D. Petri de Fermat, Toulouse, 1679; reprinted by Culture et Civilization, Brussels, 1969; pp. 173 176. The version in the Oeuvres has had its orthography modernized.

On p. 174 of the Varia (= p. 190 of the Oeuvres), he says: "j'ay trouvé une regle generale pour ranger tous les coubes à l'infiny, en telle façon que toutes les lignes de leurs quarrez tant diagonales, de largeur, de longeur, que de hauteur, fassent un méme nombre, & determiner outre cela en combien de façons differentes chaque cube doit étre rangé, ce qui est, ce me semble, une des plus belles choses de l'Arihmetique [sic]..." He describes a assembly of four squares making a magic cube. [The squares are missing in the Varia.] He says that the magic sum occurs on 72 lines, but it fails to have the magic sum on 8 of the 2 agonals and all 4 of the 3 agonals.

Lucas. Letter. Mathesis 2 (1882) 243 245. First publication of the magic 4³ described by Fermat above. Says it will appear in the Oeuvres.

E. Fourrey. Op. cit. in 4.A.1. 1899. Section 317, p. 257. Notes that Fermat's magic cube has only 64 magic lines.

Lucas. L'Arithmétique Amusante. 1895. Note IV: Section IV: Cube magique de Fermat, pp. 225 229. Reproduces the 4³ from his Mathesis letter and gives a generalization by V. Coccoz, for which the same diagonals fail to have the magic sum, though he implies they do have the sum on p. 229.

Pierre de Fermat. Letter to Mersenne, nd [Jun? 1641]. Opp. cit. above: Oeuvres, vol. 2, pp. 195 199; Varia Opera, pp. 176 178. On p. 177 of the Varia (= p. 197 of the Oeuvres), he says: "Pource qui est des cubes, je n'en sçay pas plus que Monsieur Frenicle, mais pourtant je puis les ranger tous à la charge que les Diagonales seules de quarrez que nous pouvons supposer paralleles à l'Horizon, seront égales aux côtez des quarrez, ce qui n'est pas peu de chose. En attendant qu'une plus longue meditation découvre le reste, je dresseray celuy de 8. 10. ou 12. à ces conditions si Monsieur de Frenicle me l'ordonne."

Joseph Sauveur. Construction générale des quarrés magiques. Mémoires de l'Académie Royale des Sciences (1710 (1711)) 92 138. ??NYS -- mentioned by Brooke (below), who says Sauvier [sic] presented the first magic cube but gives no reference. Discussed by Cammann 4, p. 297, who says Sauveur invented magic cubes and Latin squares. This paper contains at least the latter and an improvement on de la Hire's method for magic squares, but Cammann doesn't indicate if this contains the magic cube.

Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205. ??NX. See 4.B.1 for more details. F. 308: Essay towards forming a Magick Cube, c1840?? Very brief notes.

Gustavus Frankenstein. [No title]. Commercial (a daily paper in Cincinnati, Ohio) (11 Mar 1875). ??NYS. Perfect 8³. Described by Barnard, pp. 244 248.

Hermann Scheffler. Die magischen Figuren. Teubner, Leipzig, 1882; reprinted by Sändig, Wiesbaden, 1981. Part III: Die magische Würfel, pp. 88 101 & plates I & II, pp. 113 & 115. He wants all 2   and 3 agonals to add up to the magic constant, though he doesn't manage to construct any examples. He gives a magic 5³ which has the magic sum on 14 of the 30 2 agonals and many of the broken 2 agonals. He also gives a 4³ and a 5³, but I haven't checked how successful they are.

F. A. P. Barnard. Theory of magic squares and of magic cubes. Memoirs of the National Academy of Science 4 (1888) 209 270. ??NYS. Excerpted, including the long footnote description of Frankenstein's 8³, in Benson & Jacoby (below), pp. 32 37, with diagrams of the result on pp. 37 42.

C. Planck, on pp. 298 & 304 of Andrews, op. cit. in 4.B.1.a, says the first magic 6³ was found by W. Firth of Emmanuel College, Cambridge in 1889.

Pao Chi-shou. Pi Nai Sahn Fang Chi (Pi Nai Mountain Hut Records). Late 19C. ??NYS -- described by Lam (in op. cit. in 7.N under Yang Hui), pp. 321-322, who says it has magic cubes, spheres and tetrahedrons. See also Needham, p. 60.

V. Schlegel. ?? Bull. Soc. Math. France (1892) 97. ??NYS. First magic 3⁴. Described by Brooke (below).

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Magic cubes, pp. 99-100. 3³ magic cube with also the 6 2-dimensional diagonals through the centre having the same sum, so there are 37 lines with the magic sum. 4³ magic cube -- this has the 3 x 16 + 4 = 52 expected magic lines, but he asserts it has 68 magic lines, though I can only find 52. The perfect case would have 76 magic lines.

C. Planck. Theory of Path Nasiks. Privately printed, Rugby, 1905. ??NYS. (Planck cites this on p. 363 of Andrews, op. cit. in 4.B.1.a, and says there are copies at BM, Bodleian and Cambridge.) The smallest Nasik (= perfectly pandiagonal) kⁿ has k = 2ⁿ. If the cube is also associated, then k = 2ⁿ + 1. He quotes these results on p. 366 of Andrews and cites earlier erroneous results. On p. 370 of Andrews, he says that a perfect k⁴ has k  8.

Collins. Book of Puzzles. 1927. A magic cube, pp. 89-90. 3³, different than that in Berkeley & Rowland, but with the same properties.

J. Barkley Rosser & Robert J. Walker. MS deposited at Cornell Univ., late 1930s. ??NYS. (Cited by Gardner, loc. cit. below, and Ball, MRE, 11th ed., p. 220; 12th ed., p. 219.) Finds a Nasik 8³ and shows that Nasik k³ exist precisely for the multiples of 8 and for odd k > 8.

G. L. Watson. Note 2100: To construct a symmetrical, pandiagonal magic cube of oddly even order 2n  10. MG 33 (No. 306) (Dec 1949) 299 300.

Maxey Brooke. How to make a magic tessarack. RMM 5 (Oct 1961) 40 44. Cites Sauvier and Schlegel. Believes this is the first English exposition of Schlegel. The 3³ he develops is magic, but only the 6 2 agonals through the centre have the magic sum. The resulting 3⁴ is magic but not perfect.

Harry Langman. Play Mathematics. Hafner, 1962. ??NYS -- cited by Gardner below. Pp. 75 76 gives the earliest known perfect 7³.

Birtwistle. Math. Puzzles & Perplexities. 1971. The magic cube, pp. 157 & 199. 3³ which is associated, but just the 6 2-agonals through the centre have the magic sum; the other 12 2-agonals do not. The space diagonals also have the magic sum.

John Robert Hendricks. The third order magic cube complete. JRM 5:1 (1972) 43 50. Shows there are 4 magic 3³, inequivalent under the 48 symmetries of the cube. None of these is perfect. (The author has published many articles on magic cubes in JRM but few seem appropriate to note here.)

Gardner. 1976. Op. cit. in 7.N. Gives Richard Lewis Myers Jr.'s proof that a perfect 3³ does not exist, and Richard Schroeppel's 1972 proof that a perfect 4³ does not exist. (Gardner says Schroeppel published a memorandum on this, ??NYS.) Says that perfect cubes of edge 5, 6, 7 are unknown and gives a perfect, associated 8³ found by Myers in 1970. The Addendum in Time Travel cites Planck and Rosser & Walker for earlier 8³ and says that many readers found a perfect 7³ and refers to Langman. Also 9³, 11³ and higher orders were found.

Johannes Lehmann. Kurzweil durch Mathe. Urania Verlag, Leipzig, 1980. No. 8, pp. 61 & 160. Arrange 0 - 15 on the vertices of a 2⁴ hypercube so that each 2-dimensional face has the sum 30.

John R. Hendricks. The perfect magic cube of order 4. JRM 13 (1980 81) 204 206. Shows it does not exist.

William H. Benson & Oswald Jacoby. Magic Cubes -- New Recreations. Dover, 1981. Summarises all past results on p. 5. There are perfect n³ for n  6 (mod 12), n  7, except n = 10. It is not clear if they have proofs for n  4 (mod 8) or n  2 (mod 4). They are unable to show the non existence for n  5 (cf. p. 29). There are pandiagonal n³ for n  6 (mod 12), n  4, though it is not clear if they have a proof for n  2 (mod 4) (cf. p. 102).

Rudolf Ondrejka. Letter: The most perfect (8 x 8 x 8) magic cube? JRM 20:3 (1988) 207 209. Says Benson & Jacoby sketch a perfectly pandiagonal 8³. He gives it in detail and discusses it.

Joseph Arkin, David C. Arney & Bruce J. Porter. A perfect 4 dimensional hypercube of order 7 -- "The Cameron Cube". JRM 21:2 (1989) 81 88. This is the smallest known order in four dimensions.

Allan William Johnson Jr. Letter: Normal magic cubes of order 4M+2. JRM 21:2 (1989) 101 103. Refers to Firth, 1889, who is mentioned by Planck in Andrews. Gives a program to compute (4M+2)³ cubes and gives a 6³ in base 10 & base 6.

John Robert Hendricks. The magic tessaracts of order 3 complete. JRM 22:1 (1990) 15 26. Says there are 58 of them and gives some history.

Jacques Bouteloup. Carrés Magiques, Carrés Latins et Eulériens. Éditions du Choix, Bréançon, 1991. Nice systematic survey of this field, analysing many classic methods. Includes some material on magic cubes.
7.N.2. MAGIC TRIANGLES
There are quite a number of possible types here and I have not been very systematic in recording them.
Frenicle de Bessy. Letter to Mersenne, Mar 1640. In: Oeuvres de Fermat, op. cit. in 7.N.1, vol. 2, pp. 182 185. Discusses a magic triangle.

Scheffler. Op. cit. in 7.N.1. 1882. Part II: Das magische Polygon, pp. 47 88 & Plate I, p. 113. He considers nested n gons with the number of numbers on each edge increasing 1, 3, 5, ... or 2, 4, 6, ..., such that each edge of length k > 2 has the same sum and the diameters all have the same sum, though these sums are not all the same. He develops various techniques and gives examples up to 26 gons and 5 level pentagons.

H. F. L. Meyer. Magic Triangles. In: M. Adams; Indoor Games; 1912; pp. 357 362. He divides a triangle by lines so the triangle of order three has rows of 1, 3, 5 cells. He gets some lines of 2 and of 3 to add to the same value, and then considers hexagons of six cells, but doesn't really get anywhere.

Peano. Giochi. 1924.