7. arithmetic & number theoretic recreations a. Fibonacci numbers

Pp. 47-48, art. 20. 15 card trick.

Pp. 60-61, art. 33. 16 card trick using awkward binary.

Boy's Own Book. To tell a card thought of blindfold. 1828-2: 390-391; 1829 (US): 197; 1855 & 1859: 542-543; 1868: 640-641; 1880: 670-671; 1881 (NY): 196. 21 card trick.

Boy's Own Book. 1843 (Paris).

Pp. 396-397. "A certain number of cards being shown to a person, to guess that which he has thought of." Describes the trick with any multiple of three and says it is easier with an odd multiple such as 15, 21 or 27. Remarks that once you know the card, you can shuffle them and pick the card from the pack. c= Boy's Treasury, 1844, pp. 326-327: To guess the card thought of. = de Savigny, 1846, p. 276-277: Deviner la carte pensée. = The Secret Out, 1859, pp. 80-81: To Guess the Card Thought of.

P. 345. "To tell the number thought of by a person." PB1. = Boy's Treasury, 1844, p. 302. = de Savigny, 1846, pp. 289-290: Deviner un nombre pensé par une autre personne.

Magician's Own Book. 1857. Which counter has been thought of out of sixteen?, pp. 226 227. Like Prevost but clearer. Uses 16 cards and repeated halving to bring the thought of card to the top in four steps. Says it can be extended to 32 cards. = Boy's Own Conjuring Book, 1860, pp. 196-197.

Vinot. 1860.

Art. XVII: Deviner le nombre pensé par une personne sans lui rien demander, pp. 34-36. Do 3/2 twice, then subtract twice the original number -- this produces a + b + c. Have him then divide this repeatedly by two, taking the smaller halves -- this essentially gives you the binary digits for a + b + c, which you compute. Then form 4(a + b + c) which is x + 3a + 2b, so you can recover x by using the knowledge of when there was an odd division in the original 3/2-ings.

Art. XXIII: Deviner un nombre pensé, pp. 41-44. Premier moyen. PB1.

Art. XXXV: D'un nombre de cartes (15, 21 ou 27) deviner celle qui aura été pensée, pp. 54-55. 21 card trick clearly explained, using 21 as his example.

Indoor Tricks and Games. Success Publishing, London, nd [1930s??]. The wonderful twenty-seven & Variants of the 27-card trick, pp. 47-51. Uses ternary to locate and position a card among 27 -- but I find the description of how to perform the trick a bit cryptic.
7.M.5. LOONY LOOP = GORDIAN KNOT
This is a configuration of wire loops attached to a rod with a loop of string to be removed. The solution method uses the binary pattern of the Chinese Rings. I have a note of an early 19C version, but no details -- ??
George E. Everett, via George Barker. UK Patent 15,971 -- An Improved Puzzle. Applied: 21 Sep 1891; accepted: 24 Oct 1891. 1p + 1p diagrams. Barker states that the invention was communicated by Everett, of Grand Island, Nebraska. There is no indication of the binary pattern in the puzzle.

Though this puzzle does not appear in Hoffmann, 1893, Hordern has included it in a photo of Victorian puzzles omitted by Hoffmann on p. 256 of Hoffmann Hordern. Unfortunately, this example has no date associated with it.

It appears in Jaques puzzle boxes of c1900, named The Canoe Puzzle. I have an example and Dalgety has several examples of these boxes with the puzzle and the solution which says that if the cord gets entangled, it can be cut and replaced!

M. Adams. Indoor Games. 1912. Pp. 337 341. The double link (= Loony Loop or Satan's Rings).

Ch'ung En Yü. Ingenious Ring Puzzle Book. 1958. Op. cit. in 7.M.1. P. 21 shows a simple version.
7.M.6. BINARY BUTTON GAMES
New section.

The problems here are usually electronic puzzles with an array of lightable buttons which can take on two states -- lit and unlit. Pressing a button toggles the lights on a certain pattern of buttons. The earliest commercial example I know is the XL-25. The box says Patent No. 122-8201 061. However, there is an older similar electrical switching problem by Berlekamp -- see below. There are also mechanical versions, such as Game Jugo from Japan (mid 1980s?), which has 15 petals such that turning one over turns over some others, and Orbik -- see below. A number of further versions have appeared and I have seen some incorporated in game packages on computers. Rubik's Clock is essentially the same kind of problem except that the states of the clocks and the possible turns are (mod 12) instead of (mod 2), though the main interest centres on whether a clock is correct or not. Likewise Orbik is essentially the same, but with values (mod 4).

Elwyn Berlekamp. Switching Game. Devised and actually built at Bell Labs, c1970. This is an n x n array of lights with 2n switches which will invert the lights in a row or column. Berlekamp's version is n = 10. See Fishburn & Sloane below.

Lazlo Meero. The XL-25. Exhibited at the London Toy Fair in 1983 and marketed by Vulcan Electronics of London. The box says Patent No. 122-8201 061.

David Singmaster. The XL-25. Cubic Circular 7/8 (Summer 1985) 39-42. The XL-25 has a 5 x 5 array of buttons and two choices of toggle patterns -- a simple cross of 5 cells and the pattern of a cell and the cells which are a knight's move away. In both cases, cells off the edge are simply lost. I show that such problems with n buttons can be specified by an n x n input/output or transition binary matrix A = (a_ij), with a_ij = 1 if pressing the j-th button toggles the i-th button; otherwise a_ij = 0. If x is a binary column vector showing which buttons are pushed, then Ax is a binary column vector showing which buttons have been toggled. From a random start, the object is to light all buttons. So the general solution of the problem is obtained by inverting A (mod 2). For the 5 x 5 array, the knight version is invertible and I sketch its inverse. But the cross version gives a matrix of rank 23. I note that when the matrix is singular, the Gauss-Jordan elimination method for the inverse yields the null space and range and a kind of pseudo inverse, allowing for complete analysis -- in theory. (More recently, I have seen that this is a proper pseudo-inverse and all the pseudo-inverses of A are computable by this process.) Again I sketch the solution process. (I now incorporate this idea in my teaching of linear algebra as a handout "On trying to invert a singular matrix".) I corresponded with Meero who had obtained some similar results and showed that each feasible pattern for the cross version could be obtained in at most 15 moves. He also showed that if the input/output matrix A is symmetric and reflexive (i.e. a_ii = 1), then one can turn on all the lights, starting with them all off. He studied the cross version on n x n boards up to n = 100. A friend of Meero studied fixed point patterns such that Ax = x and anti-fixed points where Ax is the vector complementary to x. Meero asserts the knight version has A invertible if n  6, 7, 8 (mod 9). I wondered what happens for cylindrical or toroidal boards, but made no attempt to study them.

In Spring 1986, I applied this method to a Japanese puzzle called Game Jugo which Edward Hordern had shown me. This has 15 two-sided 'petals' around a centre which has four pointers. When one pointer points to petal 1, the others point to 4, 8, 11 and turning one of these over turns over the others in this set. Since the sum of all rows of the transition matrix is the zero vector, it follows that the matrix is singular. I found that it had rank 12, so there are just 2¹² = 4096 achievable patterns, each of which has 2³ = 8 solutions. It turns out that there are three groups of 5 petals, e.g. {1, 4, 7, 10, 13}, such that the sum of the turns in a group must be 0 for an achievable pattern. From this, I showed that any achievable pattern could be reached in at most 6 moves, determined how many patterns required each number of moves and showed that the average number of moves was 4.6875. I am very keen to get an example of this and/or its instructions (preferably with an English translation).

Donald H. Pelletier. Merlin's magic square. AMM 94:2 (Feb 1987) 143 150. Merlin is a product of Parker Brothers and provides several games, including a 3 x 3 binary button game. If we number the cells 1, 2, 3; 4, 5, 6; 7, 8, 9; then: pressing 1 toggles 1, 2, 4, 5; pressing 2 toggles 1, 2, 3; pressing 5 toggles 2, 4, 5, 6, 8 (same as the cross on the XL-25). The object is to light all but button 5. He develops the binary vectors as above, finds that the transition matrix is invertible, computes the inverse and answers a few simple questions.

T. E. Gantner. The game of quatrainment. MM 61:1 (Feb 1988) 29-34. Considers a game on a 4 x 4 field where a corner move reverses the six cells closest to the corner; an edge move reverse the the neighbouring cells and a centre move reverse the cell and its four neighbours (i.e. the + pattern). Sets up the matrix approach and shows the transition matrix is invertible, finding inputs which reverse just one cell. Modifies the moves and finds versions of the game with ranks 12 and 14.

Wiggs, Christopher C. & Taylor, Christopher J. C. US Patent 4,869,506 -- Logical Puzzle. Filed: 29 Jul 1988; patented: 26 Sep 1989. Cover page + 2pp + 6pp diagrams. This is the patent for what was marketed by Matchbox as Rubik's Clock in 1988. The address of Wiggs and Taylor is just across the street from Tom Kremer's firm which is Rubik's agent. This is essentially the same kind of problem except that the states of the clocks and the possible turns are (mod 12) instead of (mod 2), though the main interest centres on whether a clock is correct or not. 18 clock dials -- 9 on front and 9 on back, both in a 3 by 3 pattern. Four drive wheels on the edges, connected to the corner clocks, but their effects on other clocks are determined by the positioning of 4 buttons in the middle of the puzzle, giving 30 types of move. The four corner front clocks are connected to the four corner rear clocks, so there are 14 independent motions to make and the input/output matrix is 30 by 14.

Daniel L. Stock. Merlin's magic square revisited. AMM 96:7 (Aug/Sep 1989) 608-610. He gives an easy algorithm for solving the problem by doing edges, then corners, then middle.

P. C. Fishburn & N. J. A. Sloane. The solution to Berlekamp's switching game. Discrete Mathematics 74 (1989) 263-290. They describe Berlekamp's game, with photo, as a coding theory problem. The transition matrix A here is 2n x n². For any given initial state x₀, consider all the states that can be achieved from it, say S(x₀)  =  { x₀ + Ax | x  Z₂²ⁿ }. We might expect S(x₀) to have 2²ⁿ states, but reversing all rows is the same as reversing all columns -- and there is no other dependence -- so there are 2^2n 1 states. Among all these states, there is one with a minimal number, f(x₀), of lights turned on. The covering radius R of the code formed by the 2n rows and columns, considered as words in Z₂^{n^2}, is the maximum of these minimal numbers, i.e. min { f(x₀) | x₀  Z₂^{n^2} }. These codes are called 'light bulb' codes and have been investigated since c1970 since they have the smallest known covering radius. From our game point of view, the problem corresponds to finding the most-unsolvable position and R is a measure of unsolvability. The values for R were known for n  5. The authors use extensive hand computing to extend this up through n = 9 and then a lot of computer time to get to n = 10. The values of R for n  =  1, .., 10 are: 0, 1, 2, 4, 7, 11, 16, 22, 27, 34. That is, for the 10 x 10 game, there is an array of 34 turned-on lights which cannot be reduced to a smaller number of turned-on lights by any inversion of rows and columns.

Orbik. Orbik is a ring of 12 wheels, each having 4 colours but just one colour can be seen through the top cover of windows. There are three marks. When one mark is at 1, the others are at 4 and 8. When the top cover is turned ahead, the marked wheels move forward one colour. A backward turn leaves everything fixed, but moves the position of the marks. Made by James Dalgety. I believe it was Edward Hordern's exchange gift for a puzzle party, c1993.

Edward Hordern. Orbik. CFF 29 (Sep 1992) 26-27. ??NYR. Orbik is a ring of 12 wheels, each having 4 colours but just one colour can be seen through the top cover of windows. There are three marks. When one mark is at 1, the others are at 4 and 8. When the top cover is turned ahead, the marked wheels move forward one colour. A backward turn leaves everything fixed, but moves the position of the marks.

Ralph Gasser. Orbik. CFF 32 (Aug 1993) 26-27. He counts both forward and backward turns and finds there are 60 antipodal positions requiring 54 moves to solve. The shortest processes for moving a single wheel by 1, 2, 3 colours take 29, 28, 25 moves. If a sequence of turns in the same direction is counted as a single move, there are 4 antipodal positions requiring 23 moves to solve and the single wheel processes take 9, 7, 9 moves.

Revital Blumberg, Michael Ganor & Avish J. Weiner. US Patent 5,417,425 -- Puzzle Device. Filed: 8 Apr 1994; patented: 23 May 1995. Cover page + 3pp + 1p correction + 4pp diagrams. This is the patent for Lights Out, which is essentially identical to XL-25, but has some additional patterns. Patents 5,573,245 & 5,603,500, granted to different groups of people, continue this. No reference to Meero or any Hungarian patent, but cites Parker Brothers' Merlin as undated.

Dieter Gebhardt. Cross pattern piling. CFF 33 (Feb 1994) 14-17. Notes that Dario Uri independently invented the XL-25 idea with the cross pattern -- he called it Matrix of Lights. Gebhardt modifies the problem by making two ons remain on. Thus his computation of Ax is an ordinary matrix product and he wants results with each entry the same. If one thinks of the cross shapes as five cubes piled onto the board, the sought result is a uniformly stacked board of some height h. This also allows for some cell to be turned on several times. Thus we are trying to solve Ax = hJ, where J is the vector of all 1s, h is a positive integer and x is a vector with non-negative integer entries. Obviously the minimal value of h is wanted. He determines solvability and all minimal solutions up to 8 x 8, with 9 x 9 given as a contest.

Tiger Electronics, 980 Woodlands Parkway, Vernon Hills, Illinois, 60061, USA & Belvedere House, Victoria Avenue, Harrogate, UK. Lights Out. Model 7-574, 1995. Essentially the same concept as the XL-25 with its 'cross' pattern. With lots of preprogrammed puzzles, random puzzles and option to input your own puzzles. The longest solution is 15 moves, as found by Meero for the XL-25.

Uwe Mèffert produced Orbix (or Light Ball) in 1995 for Milton Bradley. I advised a bit on the design of the games. This is a sphere with 12 light buttons in the pattern of a dodecahedron. There are four different games. The object is to turn all lights on, but in some games, one can also get all lights off. However, only the first game is a linear transformation in the sense discussed above. The later games have rules where the effect of a button depends on whether it is lit or not and even on whether the opposite button is lit or not. Nonetheless all examples are solvable in 12 moves or less.

Edward Hordern. What's up? CFF 38 (1995) 38. ??NYR. Discusses Tiger Electronics' Lights Out.

Dieter Gebhardt & Edward Hordern. How to get the lights of "Lights Out" out. CFF 39 (1996) 20-22. ??NYR. Sketches a solution.

Edward Hordern. What's up? CFF 41 (Oct 1996) 42. Discusses Tiger Electronics' Deluxe Lights Out which has a 6 x 6 array with several options -- one can affect five lights in the form of a + or of a x; a button can have effect only if it is lit, or alternately lit/unlit.
7.N. MAGIC SQUARES
4 9 2 The 3 x 3 magic square is usually given in the form on the left.

3 5 7 We denote each of the 8 possible forms by its top row. I.e. this is the

8 1 6 492 form. All Chinese material seems to give only this form, called the

Lo Shu [Lo River Writing].

7

2 An unrelated diagram, shown on the left, is called the Ho Thu

8 3 5 4 9 diagram [River Plan]. See 7.N.5 for magic versions of this shape.

1

6

Semi magic denotes a square whose rows and columns add to the magic constant, but not the diagonals.

Pandiagonal means that the 'broken diagonals' also add to the magic constant. Lucas called these diabolic and they are also called Nasik, as they were studied by Frost, who was then living in Nasik, India.

Associated or complementary means that two cells symmetric with respect to the centre add to n² + 1.

See 7.AC.3 for related pan-digital sums.

The early history of magic squares remains rather obscure. In particular, the first four Chinese sources below are extremely vague! Cammann 4 argues that magic squares had powerful mystic meanings to the Chinese, Indians and Arabs, hence were not explicitly described in writing. However, some modern scholars doubt if the Chinese had any magic square before 10C! -- cf Gardner, 1996.

There are several surveys of some or all of the history of magic squares which I list first for later reference. These provide many more references.

17 20C material has generally been omitted, but see Bouteloup. Smith & Mikami discuss several workers in Japan, but I've omitted some of them.
SURVEYS
Wilhelm Ahrens - 1. Studien über die "magischen Quadraten" der Araber. Der Islam 7 (1917) 186 250.

Wilhelm Ahrens - 2. Die "magischen Quadrate" al-Būnī's. Der Islam 12 (1922) 157 177.

Schuyler Cammann   1. The evolution of magic squares in China. J. Amer. Oriental Soc. 80 (1960) 116 124.

Schuyler Cammann   2. The magic square of three in old Chinese philosophy and religion. History of Religions 1 (1961) 37 80. ??NYR

Schuyler Cammann   3. Old Chinese magic squares. Sinologia 7 (1962) 14 53.

Schuyler Cammann   4. Islamic and Indian magic squares I & II. History of Religions 8 (1968 69) 181 209 & 271 299.

Bibhutibhusan Datta & Avadhesh Narayan Singh. Magic squares in India. Indian J. History of Science 27:1 (1992) 51-120. All references to Datta & Singh in this section are to this paper, not their book.

Menso Folkerts. Zur Frühgeschichte der magischen Quadrate in Westeuropa. Sudhoffs Archiv 65:4 (1981) 313-338.

Heinrich Hermelink. Die ältesten magischen Quadrate höher Ordnung und ihre Bildungsweise. Sudhoffs Arch. 42 (1953) 199 217.

Lam Lay Yong. 1977. See under Yang Hui below. Her commentary surveys the history.

Needham. 1958. Pp. 55 61. See also: vol. 2, 1956, pp. 393 & 442; Vol. 5, Part IV, 1980, pp. 462-472.

Jacques Sesiano   I & II. Herstellungsverfahren magischer Quadrate aus islamischer Zeit (I) & (II). Sudhoffs Arch. 64 (1980) 187 196 & 65 (1981) 251 265.

A number of the entries in 5.A -- The Fifteen Puzzle -- refer also to magic squares. See:  Loyd (1896); Cremer (1880); Tissandier (1880 & 1880?); Cassell's (1881); Hutchison (1891).

Some entries in 5.A and here give problems of sliding the Fifteen Puzzle into a magic square. See: Dudeney (1898); Anon & Dudeney (1899); Loyd (1914); Dudeney (1917); Gordon (1988) in 5.A and Ollerenshaw & Bondi below.

POSSIBLE EARLY REFERENCES
Anon. Shu Ching. c 650. Part V, book IV, The Great Plan -- commentary and book XXII, The Testamentary Charge. IN: J. Legge, trans. The Texts of Confucianism, Translated: Part I. The Shû King, the religious portions of the Shih King, the Hsiâo King. OUP, 1879, pp. 138 139 & 239. P. 138 discusses the Lo Shu and says it does not occur. On p. 139, we see: "To [Yu], Heaven gave the Great Plan with its nine divisions ..." Various commentators, from Gan Kwo on, have asserted that this was the Lo shu which appeared on the back of a tortoise in the river Lo. Legge says there is no evidence to connect the Lo shu with the Great Plan and that the commentators are indulging in leger de plume. P. 239 mentions the River Plan.

(See also: J. Legge, trans.; The Chinese Classics, etc.; Vol. III -- Part II; Trübner, London, 1865; pp. 321 325 & 554. This gives the Chinese and the English, with extensive notes.)

At this time, the number 'nine' was used to describe the largest number and hence does not necessarily imply 3².

Anon. Lun Yu (Confucian Analects). c 5C. Book IX, Tsze Han; chap. VIII. IN: J. Legge, trans. The Chinese Classics, etc. vol. 1, Confucian Analects, The Great Learning, and the Doctrine of the Mean. Trübner, London, 1861, p. 83. = The Life and Teachings of Confucius; Trübner, London, 1869, pp. 169 170, ??NX. Also in: A. Waley; The Analects of Confucius; Allen & Unwin, London, 1949, p. 140. "The river sends forth no map."

Chuang Tzu (= Kwang Sze). The Writings of Kwang Sze. c 300. Part II, sect. VII = Book XIV, Thien Yu (The Revolution of Heaven). IN: J. Legge, trans. The Texts of Tâoism. OUP, 1891. Vol. 1, p. 346. Refers to "the nine divisions of the writing of Lo."

Anon. Ta Chuan (= Hsi Tzhu Chuan) (The Great Commentary on the I Ching [= Yi Jing]). c 300?? (Needham, vol. 2, p. 307, says c 100 and vol. 5:IV, pp. 462-463, says -2C) IN: J. Legge, trans. The Texts of Confucianism, Part II: The Yî King. OUP, 1882. Appendix III, sect. 1, chap. 12, art. 73, p. 374 & note on p. 376. [There is a 1963 Dover ed. of Legge's 1899 edition.] Also as: Part I, chapter IX -- On the Oracle. IN: The I Ching, translated by R. Wilhelm and rendered into English by C. F. Baynes, 3rd ed., 1968, Routledge and Kegan Paul, London, pp. 308 310. The text is: "The Ho gave forth the map, and the Lo the writing, of (both of) which the sages took advantage." This occurs just after paragraphs on the origin of the hexagrams and legend says the Ho Thu inspired the creation of the 8 trigrams. Legge says the original Ho Thu map was considered to be lost in the -11C and the earliest reconstruction of it was presented during the reign of Hai Zung in the Sung Dynasty (1101-1125). The I Ching is often cited but only this later commentary mentions an association of numbers with concepts. Later commentators interpret this association as referring to the Ho Thu and Lo Shu diagrams, though this is not obvious from the association -- the names were not associated with the diagrams until about the 10C -- see Xu Yiu below. See Needham, vol. 2, pp. 393 & 442 for discussion of the interpolation of the diagrams into the I Ching.)

E. John Holmyard. Alchemy. Penguin, 1957, pp. 36-38, discusses magic squares in relation to Chinese alchemy and Taoism. He says the Taoist emphasis on the number 5 is related to its being the central value of the order 3 magic square. He says this relation has been studied by H. E. Stapleton, but gives no reference. Stapleton says the square of order 3 was the ground plan of the Ming-Tang or Hall of Distinction. This Hall was used for the proclamation of monthly ordinances and the Calendar (which was partly lunar and hence of variable length). When in the Ming-Tang, the Emperor became the incarnation of the god and hence the ground plan became of major importance in Chinese alchemy. Stapleton conjectures that the original numbering of the 3 x 3 array of rooms in the Hall may not have been magic, but would have had 5 in the middle and that the magic numbering may have occurred at some time and been recognised as having special properties. Holmyard indicates the Ming-Tang arose about -1000. All in all, these statements do not agree with most of the other material in this section and it would be good to locate the work of Stapleton (??NYS), which is presumably well-known to students of ancient chemistry/alchemy.

Needham, Vol. 5:IV, 1980, pp. 462-472. Cites Stapleton on p. 462 and indicates his work is a bit cranky, but I haven't got the details yet. He goes on to discuss why 9 was so important to the Chinese. He describes the tour of the pole-star sky-god Thai I which went through the nine cells of the Lo Shu in the order: 5, 1, 2, 3, 4, 5, pause, 5, 6, 7, 8, 9, 5. His fig. 1535 shows this from a Tang encyclopedia, though this has the 2 7 6 orientation of the square. The Chinese could see Yin and Yang (= even and odd), the Four Seasons and the Five Elements, and the Nine Directions of space, all in the Lo Shu. Consequently it was not revealed to the general public until the end of the Tang (618 907). Needham then discusses the influence of the Lo Shu on Arabic alchemical thought.

Nâgârjuna. c1C. Order 4 squares, including one later called Nâgârjunîya after him, described in a MS on magic called Kakşapuţa [NOTE: ş, ţ denote s, t with underdot.], nd. ??NYS. [A. N. Singh; History of magic squares in India; Proc. ICM, 1936, 275 276. Datta & Singh.] Datta & Singh say Nâgârjuna gives several rules for forming magic squares of order 4, but all the examples given do not use consecutive values, much less the first 16 positive integers, e.g.

n-3, 1, n-6, 8; n-7, 9, n-4, 2; 6, n-8, 3, n-1; 4, n-2, 7, n-9,

which has magic constant 2n. The Nâgârjuna square is the following, with an unrelated structure and with constant 100.

30, 16, 18, 36; 10, 44, 22, 24; 32, 14, 20, 34; 28, 26, 40, 6.

There is also no reference to Nâgârjuna or his book. Can anyone provide this?

Tai the Elder. Ta Tai Li Chi (Record of Rites). c80. Chap. 67, Ming Thang. ??NYS (See Needham, p. 58.) Chap. 8, p. 43 of Szu pu ts'ung k'an edition, Shanghai, 1919 1922. Describes the 492 form. (See Cammann 2.) (Cammann 1, Lam and Hayashi say this is the first clear reference.)

Anon. I Wei Chhien Tso Tu. c1C. Chap. 2, p. 3a. ??NYS (Translated in Needham, p. 58.)

Anon. Lî Kî. c2C. Book VII -- Lî Yun, sect. IV. IN: J. Legge, trans. The Texts of Confucianism, Part III: The Li Ki, I X. OUP, 1885. Pp. 392 393. "The Ho sent forth the horse with the map (on his back) ¹. ¹ The famous 'River Map' from which, it has been fabled, Fû-hsî fashioned his eight trigrams. See vol. xvi, pp. 14-16." This last reference is ??NYS.

Theon of Smyrna. c130. Part B: Βιβλιov τα τησ εv Αριθμoσ Μoυσικησ θεωρηματα Περιεχov (Biblion ta tes en Arithmos Mousikes Theoremata Periechon). Art. 44. IN:  J. Dupuis, trans.; Théon de Smyrne; Hachette, Paris, 1892; pp. 166 169. (Greek & French.) Natural square -- often erroneously cited as magic and used to 'prove' the Greeks had the idea of magic squares.

Xu Yiu (= Hsu Yo = Xu Yue). Shu Shu Ji Yi (= Shu Shu Chi I) (Memoir on Some Traditions of Mathematical Art). 190(?). ??NYS. Ho Peng Yoke [Ancient Chinese Mathematics; IN: History of Mathematics, Proc. First Australian Conf., Monash Univ., 1980; Dept. of Math., Monash Univ., 1981, pp. 91 102], p. 94, says that this is the earliest Chinese text to give the order 3 square.

The date and authorship of this is contentious. Current belief is that this was written by Zhen Luan (= Shuzun) in c570, using the name of Xu Yue. Li & Du, pp. 96 97, say that this work first introduces the diagram. The diagram was called the "nine houses computation". The diagram was connected with the Yi Jing commentary in the 10C and then renamed Lo Shu. After the 13C, magic squares were called zong heng tu (row and column diagrams).

Needham, vol. 5:IV, p. 464, considers this as being c190, referring the Chen Luan as a commentator on it. He calls the diagram "Nine Hall computing method".

Varahamihira (= Varāhamihira (II)). Bŗhatsamhitā [NOTE: ŗ denotes r with an underdot it and the m should have an underdot.]. c550. Hayashi, below, cites a Sanskrit edition (NYS) and the following.

M. Ramakrishna Bhat. Varāhamihira's Bŗhat Samhitā [NOTE: ŗ denotes r with an underdot it and the m should have an overdot.] with English Translation, Exhaustive Notes and Literary Comments. 2 vols, Motilal Banarsidass, Delhi, 1981 1982. Vol. II: Chapter LXXVII -- Preparation of perfumes, pp. 704-718. On pp. 714-715 is the description of a 4 x 4 array:

2, 3, 5, 8; 5, 8, 2, 3; 4, 1, 7, 6; 7, 3, 6, 1, with some cryptic observations that any mixture totalling 18 is permitted, e.g. "by combining the four corners, or four things in each corner, or the central four columns, or the four central ones on the four sides." As given, many of the groups indicated do not add up to 18, nor do the columns. However, the following article notes that the bottom row should read 7, 6, 4, 1!! (Datta & Singh have this correct.)

This material is described and analysed in: Takao Hayashi; Varāhamihira's pandiagonal magic square of the order four; HM 14 (1987) 159-166. He gives the book's name as Bŗhatsamhitā [NOTE: ŗ denotes r with an underdot it and the m should have an underdot.] and says the material is in Chapter 76 (Combinations of perfumes). He gives the correct form of the array. He notes that the array is a pandiagonal magic square with constant 18, except the entries are 1, ..., 8 repeated twice. Hayashi believes that Varāhamihira must have known one of the actual magic squares which yield this square when the numbers are taken (mod 8). He shows there are only 4 such magic squares, two of which are pandiagonal. One of the pandiagonal squares is a rotation of:

8, 11, 14, 1; 13, 2, 7, 12; 3, 16, 9, 6; 10, 5, 4, 15,

which he describes as the most famous Islamic square of order 4 described in Ahrens 2. Hayashi feels that order 4 squares must originate in India, contrary to Cammann's thesis. (See also Singh, op. cit. at Nâgârjuna, above, and Ikhwān al- Şafā’ [NOTE: Ş denotes S with an underdot.], below.) Datta & Singh just say that Varāhamihira gives a magic square.)

Datta & Singh give the square beginning 2, 3, 5, 8, and says that it is a special case of

n-7, 3, n-4, 8; 5, n-1, 2, n-6; 4, n-8, 7, n-3; n-2, 6, n-5, 1.

Taking n = 9 gives Varāhamihira's square and taking n = 17 gives the square with 8, 11, 14, 1 as right hand column. One can also take the other set of the first eight integers as given, getting a square starting 2, n-6, 5, n-1. Varāhamihira's square has many magic properties and he called it 'Sarvatobhadra' (Magic in all respects). They say these properties are fully described by Varāhamihira's commentator Bhaţţotpala [NOTE: ţ denotes t with an underdot.] in 966.

Jabir ibn Hayyan (= Jâbir ibn Hayyân = Geber) (attrib.). Kitâb al Mawâzin (Book of the Balances). c800. ??NYS -- discussed in Ahrens-1. The Arabic and a French translation are in: M. Berthelot; La Chimie au Moyen Age: Vol. 3 -- "L'Alchimie Arabe"; Imprimerie Nationale, Paris, 1893. The text is discussed on pp. 19-20, where he refers to the magic square of Apollonius, with a footnote saying 'De Tyane'. The text is given on p. 118 (Arabic section) & 150 (French section). Gives 3 x 3 square in form 492.

"Here is a figure divided into three compartments, along the length and along the width. Each line of cells gives the number 15 in all directions. Apollonius affirms this is a magic tableau formed of nine cells. If you draw this figure on two pieces of linen [or rags], which have never been touched by water, and which you place under the feet of a woman, who is experiencing difficulty in childbirth, the delivery will occur immediately."

The French is also in Ahrens-1, who notes that the square does not appear in the few extant writings of Apollonius of Tyana (c100).

Hermelink mentions this as the earliest Arabic square, but gives no details. Needham, vol. 5:IV, p. 463, says Cammann gave this, and dates this as c900. Folkerts gives this as the first Arabic example.

Suter, pp. 3 4, doesn't mention magic squares for ibn Hayyan, but this appearance is simply in a list of questions on properties of animals, vegetables and minerals, so hardly counts as mathematics.

Holmyard [op. cit. above, pp. 74-75] discusses the work of Kraus and Stapleton on Jabir. Jabir considers the numbers 1, 3, 5, 8 as of great importance -- these are the entries in the lower left 2 x 2 part of his magic square. These add to 17 and everything in the world is governed by this number! He also attaches importance to 28 which is the sum of the other entries. Holmyard asserts this magic square was known to the Neo-Platonists of 3C -- an assertion which I have not seen elsewhere. Jabir uses ratios 1/3 and 5/8 extensively in his alchemical theories.

‘Ali ibn Sahl Rabbān al-Tabarī (d. 860). Paradise of Wisdom. This is a gynaecological text discovered by Siggel. ??NYS -- described in Needham, vol. 5:IV, p. 463. Example of a magic square used as a charm in cases of difficult labour. Needham thinks this is the earliest Arabic magic square.

Tâbit ibn Qorra (= Thâbit ibn Qurra). c875. This is the first reference to magic squares in Suter, on pp. 34 38, but he seems to say that the work has not survived and Ahrens-1 confirms this. Needham, vol. 5:IV, p. 463, says Cammann wonders if this ever existed.

Ikhwān al Şafā’ [NOTE: Ş denotes S with an underdot.]. Rasā’il (Encyclopedia) (??*). c983. Cairo edition, 1928, p. 69. ??NYS. Paris MS Arabe 2304 (formerly 1005) of this is the work translated by F. Dieterici as: Die Propaedeutik der Araber im zehnten Jahrhundert; Mittler & Sohn, Berlin, 1865; reprinted as vol. 3 of F. Dieterici; Die Philosophie bei den Araben im X.Jahrhundert n. Chr.; Olms, Hildesheim, 1969. Pp. 42 44 (of the 1969 ed.) shows squares of orders 3, 4, 5 and 6. The order 3 square is in the form 276. The text refers to orders 7, 8 and 9 and gives their constants. On p. 44, the translator notes that the Arabic text has some further incomplete diagrams which are not understandable. Hermelink and Cammann 4 say that the Cairo ed. is the only version to give these diagrams. Ahrens-1 says it continues with a cryptic description of the use of a 9 x 9 square on two sherds, which have not been sprinkled with water, for easing childbirth. The prescription has several more details than ibn Hayyan's.

The Arabic text and rough translation are given in: van der Linde; Geschichte und Literatur des Schachspiels; op. cit. in 5.F.1, vol. 1, p. 203. This is a description of the 3 x 3 magic square, form 492 or 294, in terms of chess moves. Ahrens-1 says that Ruska tells him that much, if not all, of the magic square and adjacent material in Dieterici was added later to the Encyclopedia. Ruska says there are many errors in the translation and Ahrens cites several further errors in nearby material.

Hermelink describes the methods and reconstructs the squares of orders 7, 8, 9 from the 1928 Cairo ed. and van der Linde. Cammann 4 says he obtained the same squares independently, but he doesn't agree on all the interpretations. He feels there are Chinese influences, possibly via India, and gives his interpretations.

The square of order 4 is given by Hayashi, op. cit. above at Varāhamihira, as:

4, 14, 15, 1; 9, 7, 6, 12; 5, 11, 10, 8; 16, 2, 3, 13. This is not pandiagonal. The square of order 7 is doubly bordered -- the first such.

(Abû ‘Alî el Hasan ibn el Hasan) (the Hs should have dots under them) ibn el Haitam. c1000. ??NYS. Suter, p. 93, says he wrote: Über die Zahlen des magischen Quadrates. He cites Woepcke, ??NYS, for MS details. Ahrens-1 indicates that the work does not exist.

J. H. Rivett Carnac. Magic squares in India. Notes and Queries (Aug 1917) 383. Quoted in: Bull. Amer. Math. Soc. 24 (1917) 106, which is cited by: F. Cajori; History of Mathematics; op. cit. in 7.L.1, pp. 92 93. The square is in the ruins of a Hindu temple at Dudhai, Jhansi, attributed to the 11C. It is 4 x 4, and each 2 x 2 subsquare also adds to 34, but the full square is not given.

Cammann 4, p. 273, says this is the same as the Jaina square at Khajuraho described below and cites the archaeological report, ??NYS. He is dubious about the date.

(Muhammed ibn Muhammed ibn Muhammed, Abû Hâmid,) el Ġazzâlî (= al Ghazzali). Mundiqh. c1100. ??NYS -- described by Ahrens-1. Ahrens cites two differing French editions which give 3 x 3 forms 492 and 294. He says the latter is a transcription error. Al Ghazzali's text is very similar to ibn Hayyan's, though one translator says the cloths are moistened. Ahrens discusses this point. He says that amulets with this magic square, called 'seal of Ghazzali' are still available in the Middle East.

Lam, p. 318, cites this as an early Arabic magic square, but doesn't give details. Suter, p. 112, doesn't mention magic squares.

Abraham ibn Ezra. Sepher Ha Schem (Book of Names), 12C, and Jesod Mora, 1158. ??NYS -- both are described in: M. Steinschneider; op. cit. in 7.B and excerpted in the next item. The material is art. 13, pp. 95++. The 672 form is shown on p. 98. Steinschneider, p. 98, also gives the 492 form and says it appears in Jesod Mora, described on pp. 99 101. Ahrens-1 only mentions that Sepher Ha Schem gives an order 3 square.

Abraham ibn Ezra. Sêfer ha Echad. c1150. Translated and annotated by Ernest Müller as: Buch der Einheit; Welt-Verlag, Berlin, 1921, with excerpts from: Jessod Mora, Sefer ha Schem, Sefer ha-Mispar and his Bible commentary.