Sources page biographical material

6.P.1. PARADOXICAL DISSECTIONS OF THE CHESSBOARD BASED

ON FIBONACCI NUMBERS
Area 63 version: AWGL, Dexter, Escott, White, Loyd, Ahrens, Loyd Jr., Ransom.
(W. Leybourn. Pleasure with Profit. 1694. ?? I cannot recall the source of this reference and think it may be an error. I have examined the book and find nothing relevant in it.)

Loyd. Cyclopedia, 1914, pp. 288 & 378. 8 x 8 to 5 x 13 and to an area of 63. Asserts Loyd presented the first of these in 1858. Cf Loyd Jr, below.

O. Schlömilch. Ein geometrisches Paradoxon. Z. Math. Phys., 13 (1868) 162. 8 x 8 to 5 x 13. (This article is only signed Schl. Weaver, below, says this is Schlömilch, and this seems right as he was a co editor at the time. Coxeter (SM 19 (1953) 135 143) says it is V. Schlegel, apparently confusing it with the article below.) Doesn't give any explanation, leaving it as a student exercise.

F. J. Riecke. Op. cit. in 4.A.1. Vol. 3, 1873. Art. 16: Ein geometrisches Paradoxon. Quotes Schlömilch and explains the paradox.

G. H. Darwin. Messenger of Mathematics 6 (1877) 87. 8 x 8 to 5 x 13 and generalizations.

V. Schlegel. Verallgemeinerung eines geometrischen Paradoxons. Z. Math. Phys. 24 (1879) 123 128 & Plate I. 8 x 8 to 5 x 13 and generalizations.

Mittenzwey. 1880. Prob. 299, pp. 54 & 105; 1895?: 332, pp. 58 & 106-107; 1917: 332, pp. 53 & 101. 8 x 8 to 5 x 13. Clear explanation.

The Boy's Own Paper. No. 109, vol. III (12 Feb 1881) 327. A puzzle. 8 x 8 to 5 x 13 without answer.

Richard A. Proctor. Some puzzles. Knowledge 9 (Aug 1886) 305-306. "We suppose all the readers ... know this old puzzle." Describes and explains 8 x 8 to 5 x 13. Gives a different method of cutting so that each rectangle has half the error -- several typographical errors.

Richard A. Proctor. The sixty-four sixty-five puzzle. Knowledge 9 (Oct 1886) 360-361. Corrects the above and explains it in more detail.

Will Shortz has a puzzle trade card with the 8 x 8 to 5 x 13 version, c1889.

Ball. MRE, 1st ed., 1892, pp. 34 36. 8 x 8 to 5 x 13 and generalizations. Cites Darwin and describes the examples in Ozanam-Hutton (see Ozanam-Montucla in 6.P.2). In the 5th ed., 1911, p. 53, he changes the Darwin reference to Schlömilch. In the 7th ed., 1917, he only cites the Ozanam-Hutton examples.

Clark. Mental Nuts. 1897, no. 33; 1904, no. 41; 1916, no. 43. Four peculiar drawings. 8 x 8 to 5 x 13.

Carroll-Collingwood. 1899. Pp. 316-317 (Collins: 231 and/or 232 (lacking in my copy)) = Carroll-Wakeling II, prob. 7: A geometrical paradox, pp. 12 & 7. 8 x 8 to 5 x 13. Carroll may have stated this as early as 1888. Wakeling says the papers among which this was found on Carroll's death are now in the Parrish Collection at Princeton University and suggests Schlömilch as the earliest version.

AWGL (Paris). L'Echiquier Fantastique. c1900. Wooden puzzle of 8 x 8 to 5 x 13 and to area 63. ??NYS -- described in S&B, p. 144.

Walter Dexter. Some postcard puzzles. Boy's Own Paper (14 Dec 1901) 174 175. 8 x 8 to 5 x 13 and to area 63.

C. A. Laisant. Initiation Mathématique. Georg, Geneva & Hachette, Paris, 1906. Chap. 63: Un paradoxe: 64 = 65, pp. 150-152.

Wm. F. White. In the mazes of mathematics. A series of perplexing puzzles. III. Geometric puzzles. The Open Court 21 (1907) 241 244. Shows 8 x 8 to 5 x 13 and a two piece 11 x 13 to area 145.

E. B. Escott. Geometric puzzles. The Open Court 21 (1907) 502 505. Shows 8 x 8 to area 63 and discusses the connection with Fibonacci numbers.

William F. White. Op. cit. in 5.E. 1908. Geometric puzzles, pp. 109 117. Partly based on above two articles. Gives 8 x 8 to 5 x 13 and to area 63. Gives an extension which turns 12 x 12 into 8 x 18 and into area 144, but turns 23 x 23 into 16 x 33 and into area 145. Shows a puzzle of Loyd: three piece 8 x 8 into 7 x 9.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. 5 x 5 into four pieces that make a 3 x 8.

M. Adams. Indoor Games. 1912. Is 64 equal to 65? Pp. 345-346 with fig. on p. 344.

Loyd. Cyclopedia. 1914. See entry at 1858.

W. Ahrens. Mathematische Spiele. Teubner, Leipzig. 3rd ed., 1916, pp. 94 95 & 111 112. The 4th ed., 1919, and 5th ed., 1927, are identical with the 3rd ed., but on different pages: pp. 101 102 & pp. 118 119. Art. X. 65 = 64 = 63 gives 8 x 8 to 5 x 13 and to area 63. The area 63 case does not appear in the 2nd ed., 1911, which has Art. V. 64 = 65, pp. 107 & 118 119 and this material is not in the 1st ed. of 1907.

Tom Tit?? In Knott, 1918, but I can't find it in Tom Tit. No. 3: The square and the rectangle: 64 = 65!, pp. 15-16. Clearly explained.

Hummerston. Fun, Mirth & Mystery. 1924. A puzzling paradox, pp. 44 & 185. Usual 8 x 8 to 5 x 13, but he erases the chessboard lines except for the cells the cuts pass through, so one way has 16 cells, the other way has 17 cells. Reasonable explanation.

Collins. Book of Puzzles. 1927. A paradoxical puzzle, pp. 4-5. 8 x 8 to 5 x 13. Shades the unit cells that the lines pass through and sees that one way has 16 cells, the other way has 17 cells, but gives only a vague explanation.

Loyd Jr. SLAHP. 1928. A paradoxical puzzle, pp. 19 20 & 90. Gives 8 x 8 to 5 x 13. "I have discovered a companion piece ..." and gives the 8 x 8 to area 63 version. But cf AWGL, Dexter, etc. above.

W. Weaver. Lewis Carroll and a geometrical paradox. AMM 45 (1938) 234 236. Describes unpublished work in which Carroll obtained (in some way) the generalizations of the 8 x 8 to 5 x 13 in about 1890 1893. Weaver fills in the elementary missing arguments.

W. R. Ransom, proposer; H. W. Eves, solver. Problem E468. AMM 48 (1941) 266 & 49 (1942) 122 123. Generalization of the 8 x 8 to area 63 version.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 23: Summat for nowt?, pp. 27-28. 8 x 8 to 5 x 13, clearly drawn.

Warren Weaver. Lewis Carroll: Mathematician. Op. cit. in 1. 1956. Brief mention of 8 x 8 to 5 x 13. John B. Irwin's letter gives generalizations to other consecutive triples of Fibonacci numbers (though he doesn't call them that). Weaver's response cites his 1938 article, above.