(1) SPSS, Order 21; pdf order 21 spss spsso21.pdf

The search for the Perfect Square with the smallest number of distinct squares (the lowest order) had been ongoing since the late 1930's.

From Duijvestijn's 1978 paper [5] reporting the discovery;

"This note is to report the existence of a simple perfect dissection of a square into 21 unequal squares.";

"The dissection was found in the night of March 22, 1978 with the aid of the DEC-10 computer of the Technological University Twente, The Netherlands. Since no simple perfect squared squares were found of orders less than 21, it is a simple perfect squared square of lowest order . Also, it is the only simple perfect squared square of order 21. So far, the lowest order simple perfect squares known, are of order 25, the first one of which, due to Wilson, was published in [4]. In total 5 simple perfect squarings of order 25 were published in Wilson’s thesis [4]. Later another 3 simple perfect squarings of order 25 were obtained by Federico [2]. The lowest order compound perfect square is still the single 24 order perfect square found by Willcocks [10] in 1948. In my thesis [8] the investigation of all 3-connected graphs of orders up to and including 20 was reported. No perfect squarings were found at that time. The Bouwkampcode [9] of the present squaring reads as follows; (50, 35, 27) (8, 19) (15, 17, 11) (6, 24) (29, 25, 9,2) (7, 18) (16) (42) (4, 37) (33). The dissection was obtained from a 3-connected planar graph of order 22 with complexity 75264. The reduction factor is 336. The reduced side of the square is 112.[5]"

  1. A. J. W. DUIJVESTIJN, A Lowest Order Simple Perfect 2 x 1 Squared Rectangle, J. Combinatorial Theory B 26, (1979), 372-374
  2. P. J. FEDERICO, private communication; Squaring rectangles and squares. A historical review with annotated bibliography, Submitted to the congress on graph theory held in July 1978 on the occasion of the 60th anniversary of Professor Tutte.
  3. P.J. FEDERICO. Note on some low-order perfect squared squares. Canad. J. Math. , 15:350–362, 1963.
  4. J. C. WILSON, “A Method for Finding Simple Perfect Squared Squarings,” Thesis, University of Waterloo, 1967.
  5. A. J. W. DUIJVESTIJN, Simple perfect squared square of lowest order, J. Combinatorial Theory B 25 (1978), 240-243.
  6. A. J. W. DUIJVESTIJN, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-336.
  7. A. J. W. DUIJVESTIJN, Two simple perfect squares of order 22, Memorandum No. 230, Twente University of Technology, September, 1978.
  8. A. J. W. DUIJVESTIJN, “Electronic Computation of Squared Rectangles,” Thesis, Technological University, Eindhoven, The Netherlands, 1962; Philips Research Reports 17 (1962), 523-612.
  9. C. J. BOUWKAMP, On the dissection of rectangles into squares, Proc. Kon. Ned. Akad. Wetensch. Amsterdam 49 (1946), 1176-1188; 50 (1947), 58-71, 72-78.
  10. T. H. WILLCOCKS, Problem 7795 and solution, Fairy Chess Rev. 7 (1948), 106.
  11. R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T.Tutte 'The Dissection of Rectangles into Squares', Duke Math. J. 7 (1940) 312–340
  12. W. T. TUTTE, Squared rectangles, pp. 3-9 in “Proceedings of the IBM Scientific Symposium on Combinatorial Problems, held on March 16-18, 1964, at the Thomas J. Watson Research Center, Yorktown Hights, N.Y.”
  13. N.D. KAZARINOFF and R. WEITZENKAMP. On the existence of compound squared squares of small order. Journal of Combinatorial Theory B , 14:163–179, 1973
  14. P. LEEUW. Compound Squared Squares. Bachelor’s thesis, Technological University Twente, August 1979.
  15. P. LEEUW. Compound squared squares : programs. Technische Hogeschool Twente, Onderafdeling der Toegepaste Wiskunde, 1980.
  16. A.J.W. DUIJVESTIJN, P.J. FEDERICO, and P. LEEUW. Compound perfect squares. The American Mathematical Monthly, 89(1):15–32, January 1982.