A014530 List of sizes of squares occurring in lowest order example of a perfect squared square.

2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42, 50

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full

Comments

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does. The order of a squared rectangle is the number of constituent squares. Duijvestijn's perfect square of lowest order (21) is simple. The lowest order of a compound perfect square is 24. [Geoffrey H. Morley, Oct 17 2012]

See the MathWorld link for an explanation of Bouwkamp code. The Bouwkamp code for the squaring is (50,35,27)(8,19)(15,17,11)(6,24)(29,25,9,2)(7,18)(16)(42)(4,37)(33). [Geoffrey H. Morley, Oct 18 2012]

Examples

			Example from _Rainer Rosenthal_, Mar 25 2021: (Start)
.
     Terms   | 2  4  6  7  8 9 11 15 16 17 18 19 24 25 27 29 33 35 37 42 50
  -------------------------------------------------------------------------
             | <-- sort selected groups
  -------------------------------------------------------------------------
  (50,35,27) | .  .  .  .  . .  .  .  .  .  .  .  .  . 27  .  . 35  .  . 50
    (8,19)   | .  .  .  .  8 .  .  .  .  .  . 19  .  .     .  .     .  .
  (15,17,11) | .  .  .  .    . 11 15  . 17  .     .  .     .  .     .  .
    (6,24)   | .  .  6  .    .        .     .    24  .     .  .     .  .
  (29,25,9,2)| 2  .     .    9        .     .       25    29  .     .  .
    (7,18)   |    .     7             .    18                 .     .  .
     (16)    |    .                  16                       .     .  .
     (42)    |    .                                           .     . 42
    (4,37)   |    4                                           .    37
     (33)    |                                               33
  _________________________________________________________________________
       Groups of terms selected and sorted for the Bouwkamp piling
.
  The Bouwkamp code says how to pile up the squares in order to tile the square with side length 50 + 35 + 27 = 112. The procedure is beautifully animated in "Eric Weisstein's World of Mathematics" entry - see link.
(End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences. Academic Press, San Diego, 1995, Fig. M4482.
• I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.

Links

• Stuart E. Anderson, Catalogues of Perfect Squared Squares
• C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992.
• C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994.
• A. J. W. Duijvestijn, Simple perfect square of lowest order, J. Combin. Theory Ser. B 25 (1978), 240-243.
• Gergely Földvári, Photo of my artwork (2022) depicting the lowest order perfect squared square using 21 distinct colors
• N. D. Kazarinoff and R. Weitzenkamp, On the existence of compound perfect squared squares of small order, J. Combin. Theory Ser. B 14 (1973).163-179. [A compound perfect squared square must contain at least 22 subsquares.]
• Trinity College Mathematical Society, The Squared Square
• Eric Weisstein's World of Mathematics, Perfect Square Dissection
• Index entries for squared squares

Crossrefs

Cf. A002839, A002962, A002881, A342558 (related by the analogy between square tilings and resistor networks).

Extensions

'Simple' removed from definition by Geoffrey H. Morley, Oct 17 2012