Sources page biographical material

5.J. SQUARED SQUARES, ETC.
NOTE. Perfect means no two squares are the same size. Compound means there is a squared subrectangle. Simple means not compound.
Dudeney. Puzzling Times at Solvamhall Castle: Lady Isabel's casket. London Mag. 7 (No. 42) (Jan 1902) 584 & 8 (No. 43) (Feb 1902) 56. = CP, prob. 40, pp. 67 & 191 193. Square into 12 unequal squares and a rectangle.

Max Dehn. Über die Zerlegung von Rechtecken in Rechtecke. Math. Annalen 57 (1903) 314 332. Long and technical. No examples. Shows sides must be parallel and commensurable.

Loyd. The patch quilt puzzle. Cyclopedia, 1914, pp. 39 & 344. = MPSL1, prob. 76, pp. 73 & 147 148. c= SLAHP: Building a patchquilt, pp. 30 & 92. 13 x 13 into 11 squares, not simple nor perfect. (Gardner, in 536, says this appeared in Loyd's "Our Puzzle Magazine", issue 1 (1907), ??NYS.)

Loyd. The darktown patch quilt party. Cyclopedia, 1914, pp. 65 & 347. 12 x 12 into 11 squares, not simple nor perfect, in two ways.

P. J. Federico. Squaring rectangles and squares -- A historical review with annotated bibliography. In: Graph Theory and Related Topics; ed. by J. A. Bondy & U. S. R. Murty; Academic Press, NY, 1979, pp. 173 196. Pp. 189 190 give the background to Moroń's work. Moroń later found the first example of Sprague but did not publish it.

Z. Moroń. O rozkładach prostokątów na kwadraty (In Polish) (On the dissection of a rectangle into squares). Przegląd Matematyczno Fizyczny (Warsaw) 3 (1925) 152 153. Decomposes rectangles into 9 and 10 unequal squares. (Translation provided by A. Mąkowski, 1p. Translation also available from M. Goldberg, ??NYS.)

M. Kraitchik. La Mathématique des Jeux, 1930, op. cit. in 4.A.2, p. 272. Gives Loyd's "Patch quilt puzzle" solution and Lusin's opinion that there is no perfect solution.

A. Schoenflies. Einführung in der analytische Geometrie der Ebene und des Raumes. 2nd ed., revised and extended by M. Dehn, Springer, Berlin, 1931. Appendix VI: Ungelöste Probleme der Analytischen Geometrie, pp. 402 411. Same results as in Dehn's 1903 paper.

Michio Abe. On the problem to cover simply and without gap the inside of a square with a finite number of squares which are all different from one another (in Japanese). Proc. Phys. Math. Soc. Japan 4 (1931) 359 366. ??NYS

Michio Abe. Same title (in English). Ibid. (3) 14 (1932) 385 387. Gives 191 x 195 rectangle into 11 squares. Shows there are squared rectangles arbitrarily close to squares.

Alfred Stöhr. Über Zerlegung von Rechtecken in inkongruente Quadrate. Schr. Math. Inst. und Inst. angew. Math. Univ. Berlin 4:5 (1939), Teubner, Leipzig, pp. 119 140. ??NYR. (This was his dissertation at the Univ. of Berlin.)

S. Chowla. The division of a rectangle into unequal squares. Math. Student 7 (1939) 69 70. Reconstructs Moroń's 9 square decomposition.

Minutes of the 203rd Meeting of the Trinity Mathematical Society (Cambridge) (13 Mar 1939). Minute Books, vol. III, pp. 244 246. Minutes of A. Stone's lecture: "Squaring the Square". Announces Brooks's example with 39 elements, side 4639, but containing a perfect subrectangle.

Minutes of the 204th Meeting of the Trinity Mathematical Society (Cambridge) (24 Apr 1939). Minute Books, vol. III, p. 248. Announcement by C. A. B. Smith that Tutte had found a perfect squared square with no perfect subrectangle.

R. Sprague. Recreation in Mathematics. Op. cit. in 4.A.1. 1963. The expanded foreword of the English edition adds comments on Dudeney's "Lady Isabel's Casket", which led to the following paper.

R. Sprague. Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate. Math. Zeitschr. 45 (1939) 607 608. First perfect squared square -- 55 elements, side 4205.

R. Sprague. Zur Abschätzung der Mindestzahl inkongruenter Quadrate, die ein gegebenes Rechteck ausfüllen. Math. Zeitschrift 46 (1940) 460 471. Tutte's 1979 commentary says this shows every rectangle with commensurable sides can be dissected into unequal squares.

A. H. Stone, proposer; M. Goldberg & W. T. Tutte, solvers. Problem E401. AMM 47:1 (Jan 1940) 48 & AMM 47:8 (Oct 1940) 570 572. Perfect squared square -- 28 elements, side 1015.

R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte. The dissection of rectangles into squares. Duke Math. J. 7 (1940) 312 340. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 10-38, with commentary by Tutte on pp. 1-9. Tutte's 1979 commentary says Smith was perplexed by the solution of Dudeney's "Lady Isabel's Casket" -- see also his 1958 article.

A. H. Stone, proposer; Michael Goldberg, solver. Problem E476. AMM 48 (1941) 405 ??NYS & 49 (1942) 198-199. An isosceles right triangle can be dissected into 6 similar figures, all of different sizes. Editorial notes say that Douglas and Starke found a different solution and that one can replace 6 by any larger number, but it is not known if 6 is the least such. Stone asks if there is any solution where the smaller triangles have no common sides.

M. Kraitchik. Mathematical Recreations, op. cit. in 4.A.2, 1943. P. 198. Shows the compound perfect squared square with 26 elements and side 608 from Brooks, et al.

C. J. Bouwkamp. On the construction of simple perfect squared squares. Konink. Neder. Akad. van Wetensch. Proc. 50 (1947) 72-78 = Indag. Math. 9 (1947) 57-63. This criticised the method of Brooks, Smith, Stone & Tutte, but was later retracted.

Brooks, Smith, Stone & Tutte. A simple perfect square. Konink. Neder. Akad. van Wetensch. Proc. 50 (1947) 1300 1301. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 99-100, with commentary by Tutte on p. 98. Bouwkamp had published several notes and was unable to make the authors' 1940 method work. Here they clarify the situation and give an example. One writer said they give details of Sprague's first example, but the example is not described as being the same as in Sprague.

W. T. Tutte. The dissection of equilateral triangles into equilateral triangles. Proc. Camb. Phil Soc. 44 (1948) 464 482. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 106-125, with commentary by Tutte on pp. 101-105.

T. H. Willcocks, proposer and solver. Problem 7795. Fairy Chess Review 7:1 (Aug 1948) 97 & 106 (misnumberings for 5 & 14). Refers to prob. 7523 -- ??NYS. Finds compound perfect squares of orders 27, 27, 28 and 24.

T. H. Willcocks. A note on some perfect squares. Canadian J. Math. 3 (1951) 304 308. Describes the result in Fairy Chess Review prob. 7795.

T. H. Willcocks. Fairy Chess Review (Feb & Jun 1951). Prob. 8972. ??NYS -- cited and described by G. P. Jelliss; Prob. 44 -- A double squaring, G&PJ 2 (No. 17) (Oct 1999) 318-319. Squares of edges 3, 5, 9, 11, 14, 19, 20, 24, 31, 33, 36, 39, 42 can be formed into a 75 x 112 rectangle in two different ways. {These are reproduced, without attribution, as Fig. 21, p. 33 of Joseph S. Madachy; Madachy's Mathematical Recreations; Dover, 1979 (this is a corrected reprint of Mathematics on Vacation, 1966, ??NYS). The 1979 ed. has an errata slip inserted for p. 33 as the description of Fig. 21 was omitted in the text, but the erratum doesn't cite a source for the result.} The G&PJ problem then poses a new problem from Willcocks involving 21 squares to be made into a rectangle in two different ways -- it is not clear if these have to be the same shape.

M. Goldberg. The squaring of developable surfaces. SM 18 (1952) 17 24. Squares cylinder, Möbius strip, cone.

W. T. Tutte. Squaring the square. Guest column for SA (Nov 1958). c= Gardner's 2nd Book, pp. 186 209. The latter = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 244-266, with a note by Tutte on p. 244, but the references have been omitted. Historical account -- cites Dudeney as the original inspiration of Smith.

R. L. Hutchings & J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35. Prob. G. Assemble squares of sides 2, 5, 7, 9, 16, 25, 28, 33, 36 into a rectangle. The rectangle is 69 x 61 and is not either of Moroń's examples.

W. T. Tutte. The quest of the perfect square. AMM 72:2, part II (Feb 1965) 29-35. = Selected Papers of W. T. Tutte; Charles Babbage Research Institute, St. Pierre, Manitoba, 1979; pp. 432-438, with brief commentary by Tutte on p. 431. General survey, updating his 1958 survey.

Blanche Descartes [pseud. of Cedric A. B. Smith]. Division of a square into rectangles. Eureka 34 (1971) 31-35. Surveys some history and Stone's dissection of an isosceles right triangle into 6 others of different sizes (see above). Tutte has a dissection of an equilateral triangle into 15 equilateral triangles -- but some of the pieces must have the same area so we consider up and down pointing triangles as + and - areas and then all the areas are different. Author then considers dissecting a square into incongruent but equiareal rectangles. He finds it can be done in n pieces for any n  7.

A. J. W. Duijvestijn. Simple perfect squared square of lowest order. J. Combinatorial Thy. B 25 (1978) 240 243. Finds a perfect square of minimal order 21.

A. J. W. Duijvestin, P. J. Federico & P. Leeuw. Compound perfect squares. AMM 89 (1982) 15 32. Shows Willcocks' example has the smallest order for a compound perfect square and is the only example of its order, 24.