7. arithmetic & number theoretic recreations a. Fibonacci numbers

Das grosse Zauberquadrat, p. 16 & fig. 33 on plate II. 49 numbered pieces to make into a magic square.

Das kleine Zauberquadrat, p. 16 & fig. 34 on plate II. 3 x 3. Diagram is only semimagic.

Bestelmeier. 1801.

Item 441: Das arithmetische Zauber Quadrat. 9 numbered pieces to into a magic square. Diagram is shown disarranged.

Item 961.a: Das Sonderbarste aller magischen Zahlen=Quadrate. 64 numbers to form into a magic square, with various groups of four to add up to half the magic constant.

Item 961.b: Gewöhnliche Zauberquadraten von 64, 48, 36, 25, 16, 9 Zahlen.

Ozanam-Hutton. 1803. Chap. 12: Remarks, 1803: 237-240 & fig. 1, plate 4. 1814: 203-207 & corrections: 366-367 & fig. 1, plate 4 & additional plate 5. 1840: 104-105, with no figure. The 1814 corrections note that Franklin's square is only semi-magic and gives another large example -- 1840 omits this.

Rational Recreations. 1824.

Exer. 7, p. 51. Order three square in the 834 form. Calls 15 the 'product' of the entries 8, 3, 4. Cf Boy's Own Book.

Exer. 9, pp. 53-54. Natural and magic squares of order 5.

Exer. 24., p. 132. Magic square of order 10.

Unger. Arithmetische Unterhaltungen. 1838. Pp. 238-243, no. 907-926. This gives lots of straightforward exercises -- e.g. find a 5 x 5 magic square with sum 96, which he does by adding 6 1/5 to each entry of a normal example.

Young Man's Book. 1839. P. 232. The Magical Square. "The Chinese have discovered mystical letters on the back of the tortoise, which is the common magical square, making each way 15, viz." Gives the 294 form, while all Chinese forms have 492. Pp. 236-238 is a straightforward section on Magic Squares.

Boy's Own Book. 1843 (Paris).

P. 342. "The digital numbers arranged so as to give the same product, whether counted horizontally, diagonally, or perpendicularly." Order 3 magic additive square, despite the title, 834 form. = Boy's Treasury, 1844, p. 300. = de Savigny, 1846, p. 290. Cf Rational Recreations, 1824.

Pp. 342-343. "Magic squares." Constructs order 5 magic square from the order 5 natural square. = Boy's Treasury, 1844, pp. 300-301. = de Savigny, 1846, pp. 291-292.

P. 347. "The figures, up to 100, arranged so as to make 505 in each column, when counted in ten columns perpendicularly, and the same when counted in ten files horizontally." This is actually an associated magic square of order 10. = Boy's Treasury, 1844, p. 305. = de Savigny, 1846, p. 293.

Indoor & Outdoor. c1859. Part II, prob. 13: Franklin's magic square, pp. 132-133. Gives Franklin's 16 x 16 square and states some of its properties, very similar to Ozanam Hutton.

Vinot. 1860. Art. CLIV: Des carrés magiques, pp. 188-201. On p. 190, he gives an association of squares with planets, as given by Pacioli, but with the addition of: 1 -- God; 2 -- matter.

Magician's Own Book (UK version). 1871. The magic square oraculum, pp. 94-98. This shows a square of order 11 and says it is "a magic square of eleven, with one in the centre", but it is not at all magic. Initially it appears to be bordered, but it is a kind of arithmetical square. 1 is in the middle. Then 2 - 9 are wrapped around the central square, going clockwise with the 2 above the 1. Then 10 - 25 are wrapped around the central 3 x 3 area, with the 10 above the 2, etc. The 'oracle' consists of thinking of a number and consulting a list of fortunes, so the 'magic square' is never used!

Carroll-Wakeling. c1890? Prob. 8: Magic postal square, pp. 10-11 & 65. The first nine values of postage stamps in Carroll's time had values 1, 2, 3, 4, 5, 6, 7, 8, 10 in units of half-pence. But the total of the values in a magic square is three times the magic constant, and these values add up to 46. So Carroll allows one of the values to be repeated and the ten values now have to be placed to make a 3 x 3 magic square. Surprisingly, the value to be repeated is uniquely determined and there is just one such square.

T. Squire Barrett. The magic square of four. Knowledge 14 (Mar 1891) 45-47 & Letter (Apr 1891) 71 & Letter (Aug 1891) 156. Says he hasn't seen Frenicle's list. Classifies the 4 x 4 squares into 12 types, and obtains 880 squares, but doesn't guarantee to have found all of them. The first letter notes that he erred in counting one type, getting 48 too many, but a friend has found 16 more. Second letter notes that the missing squares have been found by another correspondent who has compared them with Frenicle's list and then Barrett corrects some mistakes. The other correspondent notes that Frenicle had proceeded by a trial and error method and probably had found all examples.

Ball. MRE, 1st ed., 1892, pp. 108-121: Chap. V: Magic squares. Later editions amplify this material, but the material is too detailed and too repetitive to appeal to me at the moment.

Hoffmann. 1893. Chap. IV, pp. 146 & 183 191 = Hoffmann-Hordern, pp. 114-118, with photos on pp. 133 & 143.

No. 7: A simple magic square. 3 x 3 case. Photo on p. 133 shows a wood circular board with a 3 x 3 array of holes and nine numbered pegs, by Jaques & Son, registered 1858. The Hordern Collection of Hoffmann Puzzles, p. 69, shows the same puzzle, dated 1860-1890.

No. 8: The "thirty four" puzzle. 4 x 4 case. Asserts that Heywood of Manchester publish a booklet, 'The Curiosities of the Thirty four Puzzle' which has instructions for obtaining all the solutions (i.e. of the 4 x 4 magic square), ??NYS. Photo on p. 143 shows several ordinary fifteen puzzles and two definite thirty-four puzzles and one possible. At the upper right is a box with "The Great American Puzzle 9 15 & 34 3 Games in One." -- I don't know what the game involving 9 can be -- ?? Below this is a solution sheet headed "Novel and Exasperating Yankee Puzzles, 15 and 34." -- The Hordern Collection of Hoffmann Puzzles, 70, shows this with the box which is to the lower left in this photo, which reads "Perry & C^o's Calculator Puzzles. Two Games in One. The third, possible, example is at the lower right and the box just has "Number Puzzle", by McLoughlin Bros N.Y. These are all dated 1879-1885. Hordern Collection, p. 70, shows the instructions and the Perry example, dated 1880-1900.