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

1,1

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]

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

Table of n, a(n) for n=1..21.

The Trinity Mathematical Society logo

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.

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.]

Eric Weisstein's World of Mathematics, Perfect Square Dissection

Index entries for squared squares

Example

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 World of Mathematics (see link section).
(End)

Crossrefs

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

Sequence in context: A050481 A285416 A284958 * A268445 A053663 A376108

Adjacent sequences: A014527 A014528 A014529 * A014531 A014532 A014533

Keyword

nonn,fini,full

Author

N. J. A. Sloane

Extensions

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

Status

approved