Sources page biographical material

5.I. LATIN SQUARES AND EULER SQUARES
This topic ties in with certain tournament problems but I have not covered them. See also Hoffmann and Loughlin & Flood in 5.A.2 for examples of two orthogonal 3 x 3 Latin squares. The derangement problems in 5.K.2 give Latin rectangles.
Ahrens-1 & Ahrens-2. Opp. cit. in 7.N. 1917 & 1922. Ahrens-1 discusses and cites early examples of Latin squares, going back to medieval Islam (c1200), where they were used on amulets. Ahrens-2 particularly discusses work of al Buni -- see below.

(Ahmed [the h should have an underdot] ibn ‘Alî ibn Jûsuf) el Bûni, (Abû'l ‘Abbâs, el Qoresî.) = Abu l‘Abbas al Buni. (??= Muhyi'l Dîn Abû’l-‘Abbâs al Bûnî -- can't relocate my source of this form.) Sams al ma‘ârif = Shams al ma‘ârif al kubrâ = Šams al-ma‘ārif. c1200. ??NYS. Ahrens-1 describes this briefly and incorrectly. He expands and corrects this work in Ahrens-2. See 7.N for more details. Ahrens notes that a 4 x 4 magic square can be based on the pattern of two orthogonal Latin squares of order 4, and Al-Buni's work indicates knowledge of such a pattern, exemplified by the square

8, 11, 14, 1; 13, 2, 7, 12; 3, 16, 9, 6; 10, 5, 4, 15 considered (mod 4). He also has Latin squares of order 4 using letters from a name of God. He goes on to show 7 Latin squares of order 7, using the same 7 letters each time -- though four are corrupted. (Throughout, the Latin squares also have 'Latin' diagonals, i.e. the diagonals contain all the values.) These are arranged so each has a different letter in the first place. It is conjectured that these are associated with the days of the week or the planets.

Tagliente. Libro de Abaco. (1515). 1541. F. 18v. 7 x 7 Latin square with entries 1, 13, 2, 14, 3, 10, 4 cyclically shifted forward -- i.e. the second row starts 13, 2, .... This is an elaborate plate which notes that the sum of each file is 47 and has a motto: Sola Virtu la Fama Volla, but I could find no text or other reason for its appearance!

Inscription on memorial to Hannibal Bassett, d. 1708, in Meneage parish church, St. Mawgan, Cornwall. I first heard of this from Chris Abbess, who reported it in some newsletter in c1993. However, [Peter Haining; The Graveyard Wit; Frank Graham, Newcastle, 1973, p. 133] cites this as being at Cunwallow, near Helstone, Cornwall. [W. H. Howe; Everybody's Book of Epitaphs Being for the Most Part What the Living Think of the Dead; Saxon & Co., London, nd [c1895] (facsimile by Pryor Publications, Whitstable, 1995); p. 173] says it is in Gunwallow Churchyard. Spelling and punctuation vary a bit. The following gives a detailed account.

Alfred Hayman Cummings. The Churches and Antiquities of Cury & Gunwalloe, in the Lizard District, including Local Traditions. E. Marlborough & Co., London & Truro, 1875, pp. 130-131. ??NX. "It has been said that there once existed ... the curious epitaph --" and gives a considerable rearrangement of the inscription below. He continues "But this is in all probability a mistake, as repeated search has been made for it, not only by the writer, but by a former Vicar of Gunwalloe, and it could nowhere be found, while there is a plate with an inscription in the church at Mawgan, the next parish, which might be very easily the one referred to." He gives the following inscription, saying it is to Hannibal Basset, d. 1708-9. Chris Weeks was kind enough to actually go to the church of St. Winwaloe, Gunwalloe, where he found nothing, and to St. Mawgan in Meneage, a few miles away. Chris Weeks sent pictures of Gunwallowe -- the church is close to the cliff edge and it looks like there could once have been more churchyard on the other side of the church where the cliff has fallen away. In the church at St. Mawgan is the brass plate with 'the Acrostic Brass Inscription', but it is not clearly associated with a grave and I wonder if it may have been moved from Gunwallowe when a grave was eroded by the sea. It is on the left of the arch by the pulpit. I reproduce Chris Weeks' copy of the text. He has sent a photograph, but it was dark and the photo is not very clear, but one can make out the Latin square part.

Hanniball Baet here Inter'd doth lye

Who dying lives to all Eternitye

hee departed this life the 17^th of Ian

1709/8 in the 22^th year of his age ~

A lover of learning
Shall wee all dye

Wee shall dye all

all dye shall wee

dye all wee shall

The  are old style long esses. The superscript th is actually over the numeral. The 9 is over the 8 in the year and there is no stroke. This is because it was before England adopted the Gregorian calendar and so the year began on 25 Mar and was a year behind the continent between 1 Jan and 25 Mar. Correspondence of the time commonly would show 1708/9 at this time, and I have used this form for typographic convenience, but with the 9 over the 8 as on the tomb.

A word game book points out that this inscription is also palindromic!!

Richard Breen. Funny Endings. Penny Publishing, UK, 1999, p. 35. Gives the following form: Shall we all die? / We shall die all. / All die shall we? / Die all we shall and notes that it is a word palindrome and says it comes from Gunwallam [sic], near Helstone.
Joseph Sauveur. Construction générale des quarrés magiques. Mémoires de l'Académie Royale des Sciences 1710(1711) 92 138. ??NYS -- described in Cammann 4, p. 297, (see 7.N for details of Cammann) which says Sauveur invented Latin squares and describes some of his work.

Ozanam. 1725. 1725: vol. IV, prob. 29, p. 434 & fig. 35, plate 10 (12). Two 4 x 4 orthogonal squares, using A, K, Q, J of the 4 suits, but it looks like:

J, A, K, Q; Q, K, A, J; A, J, Q, K; K, Q, J, A; but the  and  look very similar. From later versions of the same diagram, it is clear that the first row should have its  and  reversed. Note the diagonals also contain all four ranks and suits. (I have a reference for this to the 1723 edition.)

Minguet. 1733. Pp. 146-148 (1864: 142-143; not noticed in other editions). Two 4 x 4 orthogonal squares, using A, K, Q, J (= As, Rey, Caballo (knight), Sota (knave)) of the 4 suits, but the Spanish suits, in descending order, are: Espadas, Bastos, Oros, Copas. The result is described but not drawn, as:

RO, AE, CC, SB; SC, CB, AO, RE; AB, RC, SE, CO; CE, SO, RB, AC;

which would translate into the more usual cards as:

K, A, Q, J; J, Q, A, K; A, K, J, Q; Q, J, K, A.

However, I'm not sure of the order of the Caballo and Sota; if they were reversed, which would interchange Q and J in the latter pattern, then both Ozanam and Minguet would have the property that each row is a cyclic shift or reversal of A, K, Q, J.

Alberti. 1747. Art. 29, p. 203 (108) & fig. 36, plate IX, opp. p. 204 (108). Two 4 x 4 orthogonal squares, figure simplified from the correct form of Ozanam, 1725.

L. Euler. Recherches sur une nouvelle espèce de Quarrés Magiques. (Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen (= Flessingue) 9 (1782) 85 239.) = Opera Omnia (1) 7 (1923) 291 392. (= Comm. Arithm. 2 (1849) 302 361.)

Manuel des Sorciers. 1825. Pp. 78-79, art. 39. ??NX Correct form of Ozanam.

The Secret Out. 1859. How to Arrange the Twelve Picture Cards and the four Aces of a Pack in four Rows, so that there will be in Neither Row two Cards of the same Value nor two of the same Suit, whether counted Horizontally or Perpendicularly, pp. 90-92. Two 4 x 4 orthogonal Latin squares, not the same as in Ozanam.

Bachet-Labosne. Problemes. 3rd ed., 1874. Supp. prob. XI, 1884: 200 202. Two 4 x 4 orthogonal squares.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles, No. XVI, pp. 17-18. Similar to Ozanam.

Hoffmann. 1893. Chap. X, no. 14: Another card puzzle, pp. 342 & 378-379 = Hoffmann Hordern, pp. 234 & 236. Two orthogonal Latin squares, but the diagonals do not contain all the suits and ranks.

A, J, Q, K; J, A, K, Q; Q, K, A, J; K, Q, J, A.

G. Tarry. Le probleme de 36 officiers. Comptes Rendus de l'Association Française pour l'Avancement de Science Naturel 1 (1900) 122 123 & 2 (1901) 170 203. ??NYS

Dudeney. Problem 521. Weekly Dispatch (1 Nov, 15 Nov, 1903) both p. 10.

H. A. Thurston. Latin squares. Eureka 9 (Apr 1947) 19-21. Survey of current knowledge.

T. G. Room. Note 2569: A new type of magic square. MG 39 (No. 330) (Dec 1955) 307. Introduces 'Room Squares'. Take the 2n(2n 1)/2 combinations from 2n symbols and insert them in a 2n 1 x 2n 1 grid so that each row and column contains all 2n symbols. There are n entries and n 1 blanks in each row and column. There is an easy solution for n = 1. n = 2 and n = 3 are impossible. Gives a solution for n = 4. This is a design for a round robin tournament with the additional constraint of 2n 1 sites such that each player plays once at each site.

Parker shows there are two orthogonal Latin squares of order 10 in 1959.

R. C. Bose & S. S. Shrikande. On the falsity of Euler's conjecture about the nonexistence of two orthogonal Latin squares of order 4t+2. Proc. Nat. Acad. Sci. (USA) 45: 5 (1959) 734 737.

Gardner. SA (Nov 1959) c= New MD, chap. 14. Describes Bose & Shrikande's work. SA cover shows a 10 x 10 counterexample in colour. Kara Lynn and David Klarner actually made a quilt of this, thereby producing a counterpane counterexample! They told me that the hardest part of the task was finding ten sufficiently contrasting colours of material.

H. Howard Frisinger. Note: The solution of a famous two-centuries-old problem: the Leonhard Euler-Latin square conjecture. HM 8 (1981) 56-60. Good survey of the history.

Jacques Bouteloup. Carrés Magiques, Carrés Latins et Eulériens. Éditions du Choix, Bréançon, 1991. Nice systematic survey of this field, analysing many classic methods. An Eulerian square is essentially two orthogonal Latin squares.