250
Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares of order 27. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 98-WSK-01). Technische Universiteit Eindhoven. Document status and date: Published: 01/01/1998 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 03. Nov. 2020

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Page 1: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Simple perfect squared squares of order 27

Citation for published version (APA):Bouwkamp, C. J. (1998). Simple perfect squared squares of order 27. (EUT report. WSK, Dept. of Mathematicsand Computing Science; Vol. 98-WSK-01). Technische Universiteit Eindhoven.

Document status and date:Published: 01/01/1998

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 03. Nov. 2020

Page 2: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing Science

Simple Perfect Squared Squares of order 27

by

C.J. Bouwkamp

EDT Report 98-WSK-01Eindhoven, July 1998

Page 3: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Department of Mathematics and Computing Science

Eindhoven University of Technology

P.O. Box 513

5600 MB Eindhoven, The Netherlands

ISSN: 0167-9708

Coden: TEUEDE

Page 4: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Simple Perfect Squared Squares of order 27

C.J .Bouwkamp

Abstract

This report is a sequel to our Catalogue of Simple Perfect Squared Squares of

orders 21 through 25 and Album of Simple Perfect Squared Squares of order

26. It lists the 227 squares of order 27 known at present. They were obtained

by computer and/or transform technique. Two open questions are emphasized.

First, how can we get all possible order-27 squares, by computer, of course?

Secondly, has transform technique, other than empiricism and trial, a mathe­

matical basis in properties of symmetry in electric network theory? For the first

we only need better computers, but solving the second problem is not that easy.

The author gives details of many transforms, their parametrization, clusters,

hoping that this may be of help in solving the second open problem.

1. Introduction

This report is a sequel to our Catalogue of Simple Perfect Squared Squares of

orders 21 through 25 and Album of Simple Perfect Squared Squares of order 26,

with 207 and 441 squares respectively. We recall that there are no other SPSS

of order less than 27. See references [1] and [2] for nomenclature, acronyms,

and the like.

Our knowledge about SPSS of orders greater than 26 is quite limited. At this

moment we have only 227 of order 27, but there might exist between one and

two thousand of them. The situation is even worse for order 28, with 234 known

squares, let alone order 29 with 226 and order 30 with 243 squares. This report

is devoted to squares of order 27.

There are two ways to discover SPSS of order 27. The older of them is by

transformation techniques due to PJF (Transforms I and II) and TRW (Trans­

forms III, IV, V, VI and VIII). The newer is AJWD's computer approach

since 1978 when AJWD found the smallest-order SPSS, now on the cover of

J.Combinatorial Theory A, B. New transforms were discovered by JDSII and

CJB, while AJWD's computer approach was continued by JDSII aswell.

To end this introduction, I want to mention two open problems: (1) What is the

Page 5: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

II

number of different SPSS of order 27, and (2) Ras transformation technique,

other than empiricism and trial, a mathematical basis in properties of symme­

try of network theory. The first problem seems beyond'the power of today's

personal computers only, but the second problem is, I think, very difficult to

solve, if possible at all. Anyway, this report will introduce the reader into many

old and new transforms, clusters of two or more squares, and cluster member­

ship of a square will be indicated if applicable. By so doing I hope to contribute

in a way to possibly solving problem (2).

2. A selection of transforms

Figure 1 shows transforms I (at left) and II (at right) due to JPF, and my own

variations la, Ib, and lIa, lIb, lIe. Especially lIe, recently discovered, is added

to the list in reference [2]. It first occurs in order-27 squares.

Figure 2 shows TRW's transform III and my own variant IlIa. Note that the

drawings are in conformity with our standard form of SPSS (element at top-left

is largest).

Figure 3 shows THW's IV and VI, followed by Figure 4 with THW's V and my

Va.

Figures 5 and 6 show TRW's VIII and my own VIlla, Vlllb, and Ville.

Transform VII of TRW is not reproduced, because we have as yet only one

example of it, and the square involved is trivially imperfect.

Figure 7 shows my own transform CJB-1, so named in Skinner's book [3], in

terms of new parameters A, Band C. Note that the octagon is just shifted

toward the left. Thus in the (electric) polar nets of the two squares, the dual

of this octagon's polar net does not occur.

Figure 8 shows CJB-4, in which the octagon rotates over a quarter turn.

Figure 9 shows CJB-5, where the hexagon is rotated and reflected.

The next four Figures (10, 11, 12, 13) show beautiful examples of a so-called

"castle-and-moat" transform, where the castle is rotated while the outer bound

of the moat is not changed. By the way, the terminology is Skinner's.

Finally, Figures 14 and 15 show transform CJB-28 and its variant CJB-28a.

In all these Figures, the letters A, B, C, ... and their combinations stand for

integers denoting the size of elemental squares. This parametrization is not

unique, sometimes not easy, because the number of parameters is always small

in comparison with the degree of freedom in drawing diagrams.

Page 6: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

iii

A-C

B-C

I

A B

~~

A-C

B-C

II

A B

-C

A-C -

B-C

Ia

A B

I C I

A-C

B-C

lIa

A B

C

A B

,..---

ClIb

A-C

B-C

A B

ICIb

N:A+B-C N+l:A+BTRANSFORM I

A-C

'-- B-C

B-C

A-C

lIe

A B

-C

Figure 1

N:A+B-C N+l:A+BTRANSFORM II

Page 7: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

IV

B

,C

A

C+2D

N:A+B+C+D

A+C+DA+B

A-D

..D

C+DA+D

N+l: 2A+B+C+D

TRANSFORM III

C

C+2D

A

D

B

N: A+B+C+D

A+C+D A+B

A-D

D

A+D C+D

N+l : 2A+B+C+D

TRANSFORM IlIa

Figure 2

Page 8: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

A

r-------,C ID

B

N: A+B+C+D

v

A+D

D

C+D

B+D

N: A+B+C+2D

TRANSFORM IV

A B

A-C C~

B-C+DC-2D

N:A+B

A+D B+D

C-DB-C+2D

D_~

C+D A-C+D

N: A+B+2D

TRANSFORM VI

Figure 3

Page 9: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

VI

B+C

TZ A-D

N: A+B+C

B+D

A-C D

N:A+B+D

TRANSFORM V

B+C

I---C A-D

N:A+B+C

B+D

A-C D

N:A+B+D

TRANSFORM Va

Figure 4

Page 10: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

N: Cx2C

A

Vll

C

2C-A-B

C-A

C-B

N+2: 3C-A-B

TRANSFORM VIII

A

B

N: Cx2C

C

2C-A-B

C-A

I

C-B

N+2: 3C-A-B

TRANSFORM VIlla

Figure 5

Page 11: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

viii

A

I B I

N: Cx2C

TRANSFORM VIIIb

2C-A-B

N+2: 3C-A-B

C

C-A

C-B

I--A- B+C

A+B+C-D A-C

B-C

BC+D

A+B-C

A+2B-2C

B-CA

A-2C A+B-D

2B-2CD

N : (A+B)x(2A+2B)

TRANSFORM VIlle

Figure 6

N+2 : 2A+3B-3C

Page 12: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

A-2B+CA+B A+2B

B

A

C

A+C A+2C A+B-2C

N: 3A+B+C

A-B+C A+CA+3B

A+B+C

C

A+3C A+B-C A+B

N : 3A+2B+2C

TRANSFORM CJB-l

Figure 7

><"

Page 13: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

20A-2B 2lA-2B

54A-6B19A-2B

22A-2B

4A/

26A-2B

20A-3B A+2B IB

2lA-B 22A+B 26A-6B 26A-4B

N : 95A-lOB

29A-2B 28A-4B

44A-4B A+2B

27A-6B

29A-3B 30A-3B

23A+B

1/ 4A B~

28A-3B 27A-3B 23A-3B 23A-B

N : lOlA-lOB

><

TRANSFORM CJB-4

Figure 8

Page 14: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

29A-2B 28A-4B

44A-4B +2B

15A-2B27A-6B

29A-3B 30A-3B

23A+B

4A B~

28A-3B 27A-3B 23A-3B 23A-B

N : lOlA-lOB

41A-4B

69A-6B

20A-2B 21A-2B

19A-2B22A-2B

2IA-B 22A+B /4A26A-6B

15A-2B

20A-3B 'A+2B 2B 26A-2B15A

N+I : llOA-IOB

TRANSFORM CJB-5

Figure 9~.

Page 15: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

2A-3B+2C\ 3C-7A+8B

d I A

5C-8A+9B I A-2BHC8A-7B-C I

es •...

4C-B

B

18A-19B-2C 5C-9A+l0BI , I 9A-8B-C

3C-6A+7B1

N : 24A- 24B+8C

TRANSFORM CJB-24

Figure 10

9A-l0B-CI

N : 24A-24B+8C

\A-4B+2C

Page 16: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

\C-SA+7B

12A-12B+3C2A-3B+2C\

8A-7B-C

4C-B

B

N : 24A- 24B+8C

I

/10A-llB-C

SC-8A+9B

18A-19B-2C

12A-12B+C

12A-12B+SC

12A-12B+7C

A-2B+4C

9A-8B-C

AV3A-4B+2C

3C-7A+8B/

~9A-I0B-C17A-18B-2C 4C

SC-9A+lOB

N+l : 24A-24B+12C g:

TRANSFORM CJB-25

Figure 11

Page 17: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

A

A-2B+4C

3C-7A+SB\

5C-9A+10BI i I

><:;:"

12A-12B+5C

12A-128+7C

5C-SA+9BlSA-19B-2C

10A-lIB-C--...,.

,....-.-,-,V3C-SA+78

2A-3B+2C,./'B

SA-78-G 4C

4C-B

9A-SB-C

12A-12B+C

\3A-48+2C

9A_I0B_C

'

17A-lS8-2C

12A-12B+3C

N : 24A..,... 24B+8C N+l : 24;A..,...24B+12C

TRANSFORM CJB-26

Figure 12

Page 18: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

A-2B+4C9A-8B-C SC-8A+9B'18A_19B_2C

A

3C-7A+8B 3A-4B+2C

10A-llB-C"",

2A-3B+2C~ h-t'3C-SA+7B

B

7A-18B-2C!+-SC-9A +lOB

8A-7B-C4C-B

N+l : 24A-24B+12C N+l : 24A-24B+12C

TRANSFORM CJB-27

Figure 13

~

Page 19: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

A A+B-E

B+C-D+E

C

N: A+B+C

A+B D

B

E A-C+D-E

N+l:A+B+D

><:::.

TRANSFORM CJB-28

Figure 14

Page 20: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

A A+B-E

B+C-D+E

C

N: A+B+C

A+BD

B

A-C+D-EE

N+l:A+B+D

TRANSFORM CJB-28a

Figure 15><~:

Page 21: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

XVlll

3. Clusters of squares

As in our Catalogue [1] and Album [2], the squares are ordered in the standard

way, here numbered from 1 to 227, as shown at bottom of page. Following lines

for Code and Elements, a third line may appear under the heading Remarks.

Since more than one third of the squares were discovered by transform tech­

niques, it is worthwhile to give details by means of clusters as defined in [2].

Drawing all possible clusters and giving parameters is impossible, for lack of

space.

The first cluster occurs at number 8, a 3-cluster with two squares of order

26, namely A19 and A31, where A (now) refers to our Album [2]. See Figure

16 for transforms I, 4, and 5. Note that Remarks does not contain details of

parametrization. For transform I the parameter list is (A,B,C)=(201,119,45);

for (CJB-)4 and (CJB-)5, this is (A,B)=(3,1).

Also shown in Figure 16 is one of the many existing 4-clusters. It is very

special, because all four castle-and-moat transforms 24 through 27 are involved

in combination with two transforms 1. The four have parameters (A,B,C) =

(81,63,44) and the two have (524,436,176). See, then, Remarks for 178 and

179, where the details are omitted. In most 4-clusters the six transforms are

different and/or have more than three parameters.

Remarks about 5-clusters is even more condensed, in that the very 10 transforms

involved are not specified. Moreover, 6-clusters, with 15 transforms, are not

mentioned at all.

In Remarks a number may be preceded by A or C, where A refers to our Album

[2], and C refers to our volume of known SPSS of order 28 (unpublished).

Otherwise, a number refers to the present table.

Transform VIII and its variants, for 1x2 rectangles [5,6] and squares are included

aswell. They form 2-clusters, or just pairs.

Finally, if some square belongs to a 5-cluster, it does not mean that it is mem­

ber of any 4-cluster. Membership is also temporal. The square could become

member of some 6-cluster if and when more squares of order 27 became known.

Page 22: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

3-cluster

4-cluster

Figure 16

XIX

Page 23: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

xx

4. References

[1] C.J .Bouwkamp and A.J.W.Duijvestijn, Catalogue of Simple Perfect Squar­

ed Squares of orders 21 through 25, EUT Report 92-WSK-03, November

1992, Eindhoven University of Technology, Eindhoven, The Netherlands.

[2] C.J.Bouwkamp and A.J.W.Duijvestijn, Album of Simple Perfect Squared

Squares of order 26, EUT Report 94-WSK-02, December 1994, Eindhoven

University of Technology, Eindhoven, The Netherlands.

[3] Jasper Dale Skinner, II, Ph.D., Squared Squares Who's Who and What's

What, published by the author, 1740 North 59th Street Lincoln, Nebraska

68505, U.S.A., ISBN 0-9636569-0-2.

[4] C.J .Bouwkamp, On some new simple perfect squared squares, Discrete

Math. 106/107 (1992) 67-75.

[5] A.J.W.Duijvestijn, Simple perfect squared squares and 2x1 squared rect­

angles of order 25, Math. Comput. 62 (1994) 325-332.

[6] A.J.W.Duijvestijn, Simple perfect squared squares and 2x1 squared rect­

angles of order 26, Math. Comput. 65 (1994) 1359-1364.

[7] C.J .Bouwkamp, On step-2 transforms for simple perfect squared squares,

Discrete Math. 179 (1998) 243-252.

Eindhoven, June 1998.

Acknowledgments

Thanks are due to Herman Willemsen for producing the originals of all figures

plus text thus far, and to Fred Simons for the originals of all squares and

comments that follow.

Page 24: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 218

62

81 75

13

6 1249f-~

20 24 3031 18 46

165 10

3326 21 43

3927 29

8070 68

AJWD

Code: (81,75,62)(13,49)(12,30,46)(31,20,24,6)(18)(16,4)(33, 43)(10,39)(26,5)(21)(27,29)(80)(70)(68)

Elements: 4,5,6, 10, 12, 13, 16, 18,20,21,24,26,27,29,30,31,33, 39,43,46,49,62,68,70,75,80,81

Page 25: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 254

131123

8

27 20 2319

'7T9li 45

52 18 [16 2268 33

15

2913

8671

5542

JDSII

Code: (131, 123) (8, 27, 20, 23,45) (68, 52, 19) (7, 9, 4) (1, 22) (5) (33, 18,2) (16) (15,86)(29,71)(55,13)(42)

Elements: 1,2,4,5, 7,8,9, 13, 15, 16, 18, 19,20,22,23,27,29,33, 42,45,52,55,68, 71,86, 123, 131

2

Page 26: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 277

129148

2533

23 1817 I9 71

56 3269

49 6

14 13 26

2915 21 18

77

60 5744 39

JDSII

Code: (148, 129)(25,33,71)(56,69,23)(17,8)(9,32)(49)(26,6)(77) (29,14,13)(1,60,21)(15)(18,57)(44)(39)

Elements: 1,6,8,9,13,14,15,17,18,21,23,25,26,29,32, 33,39,44,49,56,57,60,69,71, 77, 129, 148

3

Page 27: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 300

7895

127

17

32 6141 39

36 16 18 19 1449 27

74 4713 34 33

25 5037

11499

87

AJWD

Code: (127,95,78) (17, 61) (32, 41,39) (74, 49, 36) (2,18,19) (27,16) (14,47) (34) (33)(13,50)(25,37)(114)(99)(87)

Elements: 2, 13, 14, 16, 17, 18, 19,25,27,32,33,34,36,37,39,41,47, 49,50,61, 74, 78,87,95,99, 114, 127

4

Page 28: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 303

8796

120

44 43 2234

G2C40

1130 8

1936 37

79 54 48

49 17

56

24

119104

80

AJWD

Code: (120,87,96)(44,43)(34,22,40)(12,10)(79,30, 11)(2, 8)(37,54)(19,36)(48)(49)(56,17)(119)(104,24)(80)

Elements: 2,8, 10, 11, 12, 17, 19,22,24,30,34,36,37,40,43,44,48, 49,54,56,79,80,87,96,104,119,120

5

Page 29: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 305

99 97109

2616 43

52

7542 9

93 34

761

35

5363

2522

r-- 39

17

103

7868

56

JDSII

Code: (109,99,97) (2, 43,52) (26, 75) (93, 16) (42) (34, 9) (61) (35,7) (53,63) (103, 25)(22,39)(78)(5,17)(68)(56)

Elements: 2,5, 7, 9, 16, 17,22,25,26,34,35,39,42,43,52,53,56, 61,63,68,75,78,93,97,99, 103, 109

6

Page 30: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 318

139

179

40

99

47 52 48

72

~~ 1838 13 17

31 ~ 3012 19

16 14 11

8067 6154 56

AJWD

Code: (179,139) (40, 99) (47,52,72,48) (38, 9) (18, 30) (4,17,31) (13) (19,80) (6,12)(16, 14) (11, 67) (61) (56) (54)

Elements: 4,6,9,11,12,13,14,16,17,18, 19,30,31,38,40,47,48,52,54,56,61,67,72,80,99,139,179

7

Page 31: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 320

119

201

58 61

5564

62 67 72 12

43

20 762357 52 I13TI

~47 45

19 24

Code: (201,119) (58, 61) (55,3) (64) (62, 67, 72) (43,12) (76) (57, 5) (52,20) (23, 47, 2)(45)(13,7)(6,1)(24)(19)

EleDlents: 1,2,3,5,6,7,12,13, 19,20,23,24,43,45,47,52,55,57,58,61,62,64,67,72,76, 119,201

ReDlarks: In cluster with A19, A31. TransfofDls I, CJB-4, CJB-5.

8

CJB

Page 32: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 328

94105

129

425234

71

47 2937

82 ~.l.L- 62

35 31 32 2561

30

117 120

91

AJWD

Code: (129, 105,94) (42,52) (71, 34) (82, 47) (3, 29, 10) (37) (62) (11, 18) (35, 61, 31,32,7)(25)(30,1)(120)(117)(91)

Elements: 1,3,7,10,11,18,25,29,30,31,32,34,35,37,42,47,52, 61,62,71,82,91,94,105, 117, 120, 129

9

Page 33: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 328

151

177

6982

4759

71

1~ 21 I11 13

'9 I 7 16 56i-'-'---30 44

23

9580

53 48 52

Code: (177,151)(69,82)(71,47,59)(56,13)(15,21,11)(95)(10, 1) (16,44)(9,6)(30,7)(80)(23) (53)(4,52)(48)

Elements: 1,4,6,7,9,10,11,13,15, 16,21,23,30,44,47,48,52,53,56,59,69,71,80,82,95,151,177

Remarks: In cluster with A20, A22. Transforms I, Ib, IV.

10

CJB

Page 34: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 336

135

201

6273

34 51 3152 4768 53

21 13 922

~

16 3233 24

17

8867 75

50 56

Code: (201, 135) (62, 73) (51, 11) (68, 52, 34,47) (31, 53) (21, 13) (9, 22) (32,88) (16,33,3)(24)(75)(67,17)(56)(50)

Elements: 3,9,11,13,16, 17,21,22,24,31,32,33,34,47,50,51,52, 53,56,62,67,68,73,75,88,135,201

Remarks: In cluster with A8. Transform II.

11

CJB

Page 35: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 341

165176

24 3141 r--

47 17 7 695772 38

25 I2213 58

~

44 ~ 1920

10793

7764

Code: (176,165) (41, 24, 31, 69) (57, 72, 47) (17, 7) (38) (58) (25, 22) (44,13) (107) (3,19)(11,2)(9,93)(20)(77)(64)

Elements: 2,3,7,9,11,13,17,19,20,22,24,25,31,38,41,44,47, 57,58,64,69,72,77,93,107,165,176

Remarks: In cluster with A9. Transform I.

12

CJB

Page 36: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 350

160

190

44 40

15 76

53 51 16 17\ 363071 13120

18 4135 37 34

114

8972 75

JDSII

Code: (190,160)(44,40,76)(71,53,51, 15)(4,36)(1,17,30)(16)(20,13)(41,2)(114)(37,34)(18,35)(89)(75)(72)

Elements: 1,2,4,13,15,16,17,18,20,30,34,35,36,37,40,41,44, 51,53,71,72,75,76,89, 114, 160, 190

13

Page 37: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 364

163

201

64

9946

7283 29 35

9

37

18 I 38 32

43-----lL

25 44102

20 80 7048

f--- 8

28 36

JDSII

Code: (201,163)(64,99)(72,83,46) (29,35)(37,9)(38)(32,102)(43,18,11)(7,44,80)(25)(70)(48,20)(8,36)(28)

Elements: 7,8,9,11, 18,20,25,28,29,32,35,36,37,38,43,44,46,48,64, 70, 72,80,83,99,102,163,201

14

Page 38: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 375

94119

162

25

6944

26100

2850 58 57 56

24 30

74 6679

16 15635 -

19

8976

54

AJWD

Code: (162,94,119) (69,25) (44,100) (50, 26, 28, 58) (57, 56) (24, 2) (30) (74) (156) (66,79)(89,35,16)(3,76)(19)(54)

Elements: 2,3,16,19,24,25,26,28,30,35,44,50,54,56,57,58,66, 69,74,76,79,89,94,100, 119, 156, 162

15

Page 39: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 378

93 104

181

82

115

22l20J 33 32 38 44

7417-r4131

35 41

69

27

159

123

96

AJWD

Code: (181,93,104)(82,11)(115)(38,44)(74,22,20,33,32)(7, 13) (17,5)(12)(1,69)(6,41)(159)(35)(123,27)(96)

Elements: 1,5,6,7,11,12,13,17,20,22,27,32,33,35,38,41,44, 69,74,82,93,96,104,115,123,159,181

16

Page 40: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 381

109 103

169

41

6049 62

20 21

32 I 26

6483

89 8758

25 ~~ 23 8

123 125 133

AJWD

Code: (169,109,103) (41, 62) (60, 49) (20, 21) (11, 32, 6) (26) (83) (89, 64,87) (58) (25,16,23) (8, 133) (9, 7) (125) (123)

Elements: 6, 7, 8, 9,11, 16,20,21,23,25,26,32,41,49,58,60,62, 64,83,87,89, 103, 109,123, 125, 133, 169

Remarks: In cluster with ClO. Transform I.

17

Page 41: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 384

173

211

7796

5568

88 1913

20 ~1~ 2258

25 42

-29 27 28 24

115

85 7656 52

Code: (211, 173) (77, 96) (88, 68, 55) (58, 19) (13, 42) (115) (20, 14,22, 25) (6,8) (85,29)(27,3)(28)(24,76)(56)(52)

Elements: 3,6,8,13,14, 19,20,22,24,25,27,28,29,42,52,55,56,58,68, 76, 77,85,88,96, 115, 173, 211

Remarks: In cluster with A38. Transform 1.

18

CJB

Page 42: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 386

185201

16

6853 59 101

105

24 35 3347 41 f------t-,

17r32 34

25 30

10080 85

5566

. AJWD

Code: (201, 185)(16,68, 101)(105,53,59)(35,33)(47,6)(41,24)(17,7)(34,100)(3,32)(10)(30,85)(80,25)(66)(55)

Elements: 3,6,7,10,16, 17,24,25,30,32,33,34,35,41,47,53,55, 59,66,68,80,85,100,101,105,185,201

19

Page 43: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 391

103

134154

31

72

33 7147 94

~ 74 3042

~ 29 27 23

66 5453 55

163

108120

AJWD

Code: (154, 134, 103) (31, 72) (94, 71) (33, 47, 74) (30, 42) (19, 14) (5, 29,27) (23, 66,12)(24)(54)(55,163)(53)(120)(108)

Elements: 5,12,14,19,23,24,27,29,30,31,33,42,47,53,54,55,66, 71,72,74,94,103,108, 120, 134, 154, 163

Remarks: In cluster with C16. Transform 1.

20

Page 44: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 391

172

219

43 5376

31 3347

73 68 22Zk 63

t-- 20 4026 21 15 58

27

99 104 10385

Code: (219,172) (76, 43, 53) (33,10) (73, 47,68,31) (63) (11, 58,40) (22,9) (20) (26,21)(15,7)(27)(104)(103)(99)(85)

Elements: 7,9,10,11,15,20,21,22,26,27,31,33,40,43,47,53,58, 63,68,73,76,85,99,103, 104, 172,2] 9

Remarks: In cluster with A30. Transform II.

21

CJB

Page 45: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 403

186

217

79

48 107

7594

2846 53

24 2551 Pm 36

16110

92 8560 56

AJWD

Code: (217,186)(79, 107)(75,94,48)(46,53,28)(51,24)(25, 110)(5,36,92,7)(14,15)(85)(13,1)(16)(60,4)(56)

Elements: 1,4,5,7,13,14,15,16,24;25,28,36,46,48,51,53,56,60, 75, 79,85,92,94,107,110,186,217

22

Page 46: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 405

107

141157

34

73

5139

8569

88

63100

2945

2680 25

6236 35 67

16

12598 99

83

AJWD

Code: (157, 141, 107) (34, 73) (85, 51, 39) (88, 69) (12, 100) (63) (45,80, 29) (62, 26)(67,25)(36,35)(125)(99,16)(98)(83)

Elements: 12,16,25,26,29,34,35,36,39,45,51,62,63,67,69, 73,80,83,85,88,98,99,100, 107, 125, 141, 157

23

Page 47: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 406

195211

61 52

58 59 82

94 ~ 3020

32 r;- 4622 2136

4329

129101 104

72

AJWD

Code: (211,195)(61,52,82)(94,58,59)(9, 13,30)(20,46,4)(17)(36,22)(21,32,6)(26)(129)(43)(104)(101,29)(72)

Elements: 4,6,9, 13, 17,20,21,22,26,29,30,32,36,43,46,52,58, 59,61,72,82,94,101,104,129,195,211

24

Page 48: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 407

102

131

174

29

73

51

201

19 18 10936

16f" 4B30 58

99

63

78

28

167

134

106

AJWD

Code: (174,102,131)(73,29)(51,109)(99,36,20, 19)(18,48,7)(58) (1,6,30)(16,5)(11)(63)(78)(167)(134,28)(106)

Elements: 1,5,6,7,11,16,18,19,20,28,29,30,36,48,51,58,63, 73,78,99,102,106,109,131,134,167,174

Remarks: In cluster with C11. Transform 1.

25

Page 49: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 410

191219

4056

19

~13395

17101 99 39

22

5432

1-'--37

24 43124

90 92

55 49

JDSn

Code: (219,191) (40, 56, 95) (101, 99,19) (7, 33) (26) (17, 39) (54, 22) (32,124) (13, 37,43,92)(90,11)(24)(55,6)(49)

Elements: 6,7,11,13,17, 19,22,24,26,32,33,37,39,40,43,49,54, 55,56,90,92,95,99,101,124,191,219

26

Page 50: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 414

111131

172

4368

4818 25

83

37 29

82 RT:'":L75 ~108

38 31

8026

40

160134

120

AJWD

Code: (172, 111, 131) (43, 68) (48,83) (18, 25) (108, 82) (75,37,29) (8, 15,6) (9,80)(38,7)(31)(26,160,40)(134)(120)

Elements: 6,7,8,9,15,18,25,26,29,31,37,38,40,43,48,68,75, 80,82,83,108,111,120,131,134,160,172

27

Page 51: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 416

115

156 145

30

85

44 55

112120

4673 65

27 19

4345 53

64 60

21 22

151

8498

83

AJWD

Code: (156, 115, 145)(85,30) (55, 120)(112,44)(73,46,65) (27, 19) (53, 151) (64, 43,5)(60,45)(21,22)(98)(84,1)(83)

Elements: 1,5, 19,21,22,27,30,43,44,45,46,53,55,60,64,65, 73, 83,84,85,98, 112,115,120,145,151,156

28

Page 52: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 417

130 137150

6585 123

144

25'--- 40

3015

27 29 2880

6445

3656

23

136

102 10079

AJWD

Code: (150,130, 137)(123, 7)(144)(85,65)(25,40)(80,5)(30)(2,29,64,28)(15,27)(45)(36,136)(56)(102,23)(100)(79)

Elements: 2,5, 7, 15,23,25,27,28,29,30,36,40,45,56,64,65, 79, 80,85, 100, 102, 123,130,136,137,144,150

29

Page 53: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 419

194

225

4762

85

59 54 11532 117

112 6043

64

14030 19 1~ 29I 103

82

52 42

Code: (225,194)(85,47,62)(112,59,54)(32, 15)(17,60)(6,43)(5,140)(64)(103)(82,30)(19,16,29)(3,13)(52)(42)

Elements: 3,5,6,13,15,16,17, 19,29,30,32,42,43,47,52,54,59,60,62,64,82,85,103,112, 140, 194,225

Remarks: In cluster with A43. Transform II.

30

CJB

Page 54: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 431

107

138

186

4067 9

49H2922

129

7352 71

106

33 44 4836

164139

128

JDSII

Code: (186,107,138) (67, 40) (9,129) (49) (16, 29,22) (106, 73, 7) (3, 13) (10) (71) (52)(4,48)(33,44)(36,164)(139)(128)

Elements: 3,4, 7, 9, 10, 13, 16,22,29,33,36,40,44,48,49,52,67, 71, 73, 106, 107, 128,129, 138, 139, 164, 186

31

Page 55: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 432

109

167 156

39

70

31 1 23

67

100141

12433 34

36 35

96

72

13512927

1569

5742

JDSII

Code: (167, 109, 156) (70,39) (31, 8) (23, 141) (l00, 67) (124) (33,34) (96,36, 1) (35)(6,135)(72,129)(69,27)(15,57)(42)

Elements: 1,6,8,15,23,27,31,33,34,35,36,39,42,57,67,69, 70,72,96,100,109,124,129,135,141,156,167

Remarks: In cluster with 33, C24, C26, C28.

32

Page 56: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 433

109

168 156

39

701111

23 1 31

4269 57

27 15141

124

72

96

36 35

33 34

136 130100

67

Code: (168, 156, 109) (39, 70) (8,31) (141, 23) (69, 42, 57) (124) (27, 15) (72) (96) (36,35,136,6)(130)(1,34)(100,33)(67)

Elements: 1,6,8,15,23,27,31,33,34,35,36,39,42,57,67,69, 70,72,96,100,109,124,130, 136, 141, 156, 168

Remarks: In cluster with 32, C24, C26, C28.

33

CJB

Page 57: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 439

183

256

90 93

70 75 32 2958 64111

26 35

65 56

39 108 9972 21

33 r~ 48 52

Code: (256,183) (90, 93) (111, 70,75) (58, 32) (29,64) (26, 35) (65, 5) (56, 108) (99) (72,39)(4,52)(21,48)(33,6)(27)

Elements: 4,5,6,21,26,27,29,32,33,35,39,48,52,56,58,64,65, 70,72,75,90,93,99,108, 111, 183,256

Remarks: In cluster with A54. Transform II.

34

CJB

Page 58: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 443

211232

97114

100

132

21 Hir----'--""

5736

52 48

33 120 15

I118

79 7859 63

46

PJFffHW

Code: (232,211)(97,114)(132,100)(57,21, 10,9)(1,8)(11)(4, 118) (36)(52,48)(79,33,20)(15,78)(63)(13,59)(46)

Elements: 1,4,8,9,10,11,13,15,20,21,33,36,46,48,52,57,59,63, 78, 79,97,100,114,118, 132,211,232

Remarks: In cluster with 42, A75, A86. Transforms I, Ib, II, III, N, V.

35

Page 59: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 447

126 123

198

52 556874

57 61 948178 76

~~ 34

1743

48 53

171 175

101

Code: (198,126, 123) (55, 68) (74, 52) (94, 13) (81) (57, 61, 78, 2) (76) (43, 14) (10, 34,17) (24) (175) (171) (5, 53) (48) (101)

Elements: 2,5,10,13,14,17,24,34,43,48,52,53,55,57,61,68,74, 76,78,81,94,101,123,126, 171, 175, 198

Remarks: In cluster with C33. Transform II.

36

CJB

Page 60: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 452

203

249

80

123

6276

111 37 43

~53

39 46 4017 20

32 I 28126

92 88 8660

AJWD

Code: (249,203)(80,123)(111,76,62)(37,43)(53,9)(46) (18, 19,39) (40, 126)(17, 1)(20)(92,32,4)(28,88)(86)(60)

Elements: 1,4,9,17,18, 19,20,28,32,37,39,40,43,46,53,60,62, 76,80,86,88,92,111,123,126,203,249

37

Page 61: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 453

216237

2844 90

73 017 1269224

4819 36

5435 78

51

124 11189

69 60

AJWD

Code: (237,216)(90,126)(73,92,44,28)(11,17)(5,6)(48,1)(24) (78,36)(54,19)(35,124)(51,111)(89)(69,9)(60)

Elements: 1,5,6,9,11,17,19,24,28,35,36,44,48,51,54,60,69, 73,78,89,90,92,111,124,126,216,237

38

Page 62: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 453

201

252

84

117

6388

101 24

t"9r 6025

38 32 27 ~61

2942

16113

100 108

7458

Code: (252,201)(117,84)(88,63, 101)(24,60)(9, 15)(25,38)(32,61,27,6)(21)(113)(108)(3,29)(100,42)(16,74)(58)

Elements: 3,6,9,15, 16,21,24,25,27,29,32,38,42,58,60,61,63, 74,84,88,100,101,108,113,117,201,252

Remarks: In cluster with A58. Transform II.

39

CJB

Page 63: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 456

128145

183

17

73 11189

6923

114 18 E 5029

6045 42 34 8226

53

159 162135

JDSII

Code: (183, 145, 128) (17, 111) (89, 73) (114, 69) (23, 50) (18, 60, 11) (29,82) (4, 19)(15)(45,42)(34)(26,53)(162)(159)(135)

Elements: 4,11,15,17,18,19,23,26,29,34,42,45,50,53,60,69, 73,82,89,111,114,128,135, 145, 159, 162, 183

40

Page 64: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 456

219237

68 56

60 66 95

17111

27 r2239

3624

54 48 f--

~2841

142108 105 101

·AJWD

Code: (237,219)(68,56,95)(111,60,66)(17,39)(36,27,5)(22) (54,6)(48,24)(41,142) (4, 32) (28) (108, 3) (105) (101)

Elements: 3,4,5,6, 17,22,24,27,28,32,36,39,41,48,54,56,60,66,68,95, 101, 105,108,111,142,219,237

41

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Order 27, reduced side 458

211

247

9746 114

59 6379

I33 120 fH 21

52 48 r---57

36

132 133

100 93

PJFffHW

Code: (247,211)(114,97)(79,46,59,63)(33,13)(20,52)(9, 10,21,57)(48,15)(8,1)(11) (133, 4) (132) (36) (100) (93)

Elements: 1,4,8,9,10, 11,13, 15,20,21,33,36,46,48,52,57,59, 63,79,93,97, 100, 114,132, 133,211,247

Remarks: In cluster with 35, A75, A86. Transforms I, Ib, II, III, IV, V.

42

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Order 27, reduced side 465

240225

54

57 7794

81J102 ffsi 40

33 22 16 17

21 604351 42

146123

93 103

. JDSII

Code: (240,225) (54, 94,77) (102, 81, 57) (14, 40) (4,10) (33,22,6) (17,60) (16) (146,43) (21, 51,9) (42) (123) (103) (93)

Elements: 4,6,9,10,14,16,17,21,22,33,40,42,43,51,54,57,60, 77,81,93,94,102,103,123,146,225,240

43

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Order 27, reduced side 465

223242

97126

119 123

29

68

r-;~ 37

3928

47~0125

104 116

63 n53 52

JDSII

Code: (242,223)(97,126)(119,123)(68,29)(39,116)(104,15)(11,37, 47,28)(26)(5,25,77)(13,20)(63)(6,7)(53)(52)

Elements: 5,6,7,11,13,15,20,25,26,28,29,37,39,47,52,53,63, 68,77,97,104,116,119,123,126,223,242

Remarks: In cluster with 56, An. Transforms lb, II, V.

44

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Order 27, reduced side 467

117

154

196

25

rT-L 92

4429

106

64 62

132 103

5268

112

61

17155

139

9578

AJWD

Code: (196,154,117)(25,92)(12,13)(106,44, 15, 1)(14)(29)(132,64) (62,11)(103)(68,112,52)(155)(139,61)(17,95)(78)

Elements: 1, 11, 12, 13, 14, 15, 17,25,29,44,52,61,62,64,68, 78, 92, 95, 103, 106, 112,117, 132, 139, 154, 155,196

Remarks: In cluster with C30. Transfonn 1.

45

Page 69: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 467

218249

37 904457 128

I20F J 11124 1 34

5238

8058

25 3145

107 121

81 82 76

AJWD

Code: (249,218)(90,128)(57,37,44,111)(20,17)(10,34)(3,24)(80) (52,38)(58)(45,121)(25,107,31)(1,82)(81)(76)

Elements: 1,3, 10, 17,20,24,25,31,34,37,38,44,45,52,57,58,76, 80,81,82,90,107,111,121,128,218,249

46

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Order 27, reduced side 469

124

85

39

46

260

60

53

96 91

36

77

51

54

50

52

209

158

102

JDSII

Code: (260,209) (51, 158) (124, 96, 91) (41, 34, 16) (60, 36) (22, 50, 102) (85,39) (30,4)(26)(77)(54,2)(7,53)(52)(46)

Elements: 2,4, 7, 16,22,26,30,34,36,39,41,46,50,51,52,53,54, 60,77,85,91, 96,102,124,158,209,260

47

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Order 27, reduced side 473

225248

56 66

62 10380 l::l~106 16 20

18 25 6144 3624

5731

145119 121

88

CJB

Code: (248,225)(103,56,66)(106,62,80)(22, 19, 15)(5,61)(20)(3,16)(25)(44,18)(36) (57, 145,24) (121) (119, 31) (88)

Elemen~: 3,5,15,16,18, 19,20,22,24,25,31,36,44,56,57,61,62,66,80,88,103,106,119,121,145,225,248

Remarks: In cluster with A61. Transform II.

48

Page 72: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 483

199

284

105 94

2575 27 --C 69

102 107 t*78 31

2352

20

12197 92 101

72

PJFITHW

Code: (284, 199) (105,94) (102, 107,75) (25, 69) (78,27) (13, 12) (1, 11) (31, 10) (21)(52,23)(121)(101)(97,5)(92,20)(72)

Elements: 1,5,10,11,12,13,20,21,23,25,27,31,52,69,72, 75,78,92,94,97,101,102,105,107,121,199,284

Remarks: In cluster with A87. Transform 1.

49

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Order 27, reduced side 484

227257

106 12189 76

92

41 48 39

~7122r.-- 69

27 I 33 151136

68 63 66

AJWD

Code: (257,227)(106,121)(89,92,76)(16, 151, 15)(41,48)(136)(39,69)(29,12)(6,33)(5,22,27)(17)(68)(3,66)(63)

Elements: 3,5,6,12,15,16,17,22,27,29,33,39,41,48,63,66,68, 69,76,89,92,106,121,136,151,227,257

50

Page 74: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 490

246 244

51 ~73 36

99116 '--- 115

46 41

43 ~3:2317

I--~

44 40

147131128

84

AJWD

Code: (246,244) (8, 22, 99, 115) (116, 73,51,6) (14) (36) (46,5) (41) (43,30) (23,147,16)(13,17)(131)(128,44)(40)(84)

Elements: 5,6,8, 13, 14, 16, 17,22,23,30,36,40,41,43,44,46,51, 73,84,99, 115, 116,128,131,147,244,246

51

Page 75: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 490

226

264

92

72 13481

111

39 l:- 50 4223

5827 35 38 46

41 t-T2°1

115130

74 87 84

AJWD

Code: (264, 226) (92, 134) (111, 72, 81) (50, 42) (39, 33) (58, 23) (46, 130) (35, 38) (6,27)(115,41)(20,7)(13,87)(84)(74)

Elements: 6,7,13,20,23,27,33,35,38,39,41,42,46,50,58,72,74, 81,84,87,92,111,115,130,134,226,264

52

Page 76: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 491

215

276

61

15449 59 68 73

88

39

42 126 21

69I 52

31 3218

4025

127 122

87 9065

AJWD

Code: (276, 215) (61, 154) (88, 49, 59, 68, 73) (39, 10) (69) (42, 26) (21, 52) (127) (32,122)(16,31)(18,40)(25,90)(87)(65)

Elements: 10,16, 18,21,25,26,31,32,39,40,42,49,52,59,61,65, 68,69,73,87,88,90,122,127,154,215,276

53

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Order 27, reduced side 492

238254

104134

123 131

30

74

3244

115 107 33 41 120

~76

22 17 35

Code: (254,238)(104,134)(123,131)(74,30)(44, 120)(115,8)(107,32) (1,41,76)(33) (20, 13) (6,35) (7,12) (22, 5) (17)

Elements: 1,5,6, 7, 8, 12, 13, 17,20,22,30,32,33,35,41,44, 74, 76, 104, 107, 115, 120,123,131,134,238,254

Remarks: In cluster with 64, A108. Transforms lb, II, V.

54

CJB

Page 78: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 492

224

268

4459

34 12119

75 4087

116 53 §2226

1749 -----, '--23 20

137 147

108 100

AJWD

Code: (268,224) (44, 59,121) (116, 75, 87, 34) (19, 40) (53) (18, 22) (14, 4) (49, 26) (147)(17,137)(23,3)(20)(108,8)(100)

Elements: 3,4, 8, 14, 17, 18, 19,20,22,23,26, 34,40,44,49,53,59, 75, 87, 100, 108, 116,121,137,147,224,268

55

Page 79: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 495

223

272

97126

63 53 52

104~OJ 25

~~ 37

2947

28 6839

146119 123

107

Code: (272,223) (126, 97) (104, 63, 53, 52) (29, 68) (7,20, 25) (47, 6) (13) (26,37) (28,5) (146, 39) (15, 11) (123) (119) (107)

Elements: 5,6,7,11,13, 15,20,25,26,28,29,37,39,47,52,53,63,68,97,104, 107,119,123,126,146,223,272

Remarks: In cluster with 44, A92. Transforms lb, II, V.

56

CJB

Page 80: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 496

232264

108124

133 131

92

36 38 49 42140

99122~

63 -- 645237 41

Code: (264,232)(108,124)(133, 131)(92, 16)(140)(2,38,49,42)(99,36)(5,23,64)(63,11)(7,22,18)(52,15)(41)(37)

Elements: 2,5, 7, 11, 15, 16, 18,22,23,36,37,38,41,42,49,52,63, 64,92,99, 108, 124,131,133,140,232,264

Remarks: In cluster with Al15. Transform lb.

57

CJB

Page 81: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 503

239264

25

105 10972

114 103

31 ~8J.::,- 32 2858

4381

60

125 136101

67 14

AJWD

Code: (264,239) (25, 105, 109) (114, 103, 72) (31, 18,23) (15, 58,32) (28, 81) (13,5)(43) (11, 136) (60) (125) (101) (7, 74) (67)

Elements: 5,7,11,13,15,18,23,25,28,31,32,43,58,60,67,72,74, 81,101,103,105,109,114,125,136,239,264

58

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Order 27, reduced side 508

154140

214

60

108 126

39 5067 I--

118 28

I 374161 63

51 4422 93

56

176 183149

JDSII

Code: (214,154,140)(14,126)(60,108)(118,67,39, 50)(28, 11)(63,41,4) (61)(37,93)(51,44)(22,56)(7,183)(176)(149)

Elements: 4,7,11,14,22,28,37,39,41,44,50,51,56,60,61,63,67, 93,108,118,126,140,149,154,176,183,214

59

Page 83: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 515

244271

: 2761

34

156127 137

95

39 38 5148

123 24117 115107

59 53 64

Code: (271,244)(27,61, 156)(127, 137,34)(95)(117,10)(8,39,38,51,115)(107,48)(1,24,13)(17,23)(64)(59,6)(53)

Elements: 1,6,8,10,13, 17,23,24,27,34,38,39,48,51,53,59,61,64,95,107, 115, 117,127,137,156,244,271

Remarks: In cluster with SPSR 25:244x488. Transform VIlla.

60

CJB

Page 84: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 520

226

294

96130

61 66 77 90 3462

~1 I I56 5129 38 ~25

164136

109 111

AJWD

Code: (294,226)(96,130)(61,66, 77,90)(62,34)(56,5)(164)(51,20) (9,38,17,13)(29)(4,25,136)(21)(2,111)(109)

Elements: 2,4; 5,9,13,17,20,21,25,29,34,38,51,56,61, 62, 66, 77,90,96,109,111,130,136,164,226,294

61

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Order 27, reduced side 521

249272

41 l:- 69104 29

77 81 47 ~114 24

7440 37 33 55

43 46

168135 129

89

Code: (272,249)(104,41,35,69)(77, 114,81)(6,29)(47)(24,5)(74) (16,55)(40,37)(33,168)(3,46,135)(43)(129)(89)

Elements: 3,5,6, 16,24,29,33,35,37,40,41,43,46,47,55,69, 74, 77,81,89, 104, 114,129,135,168,249,272

Remarks: In cluster with A94. Transform II.

62

CJB

Page 86: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 524

142163

219

5231

905676

45 38

145 130121 128

36 2579

~6j 3560

160 164

96 104

cm

Code: (219, 163, 142) (52,90) (56, 76, 31) (45,38) (145, 130) (128) (121) (15, 79, 36)(25,60,164)(160)(26,35)(17,9)(104)(96)

Elements: 9,15,17,25,26,31,35,36,38,45,52,56,60,76, 79,90,96,104,121,128,130,142,145,160,163,164,219

Remarks: In cluster with C53. Transform 1.

63

Page 87: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 534

238

296

104

W-==-134

20

115 107 33 41 30

7432 44

162

123 131 118

Code: (296,238) (134,104) (115,107,22,17,35) (5,12) (20,7) (13, 6) (41) (33) (30, 74)(32,1) (162, 44) (8,131) (123) (118)

Elements: 1,5,6,7,8, 12, 13, 17,20,22,30,32,33,35,41,44,74, 104,107, 115, 118, 123,131,134,162,238,296

Remarks: In cluster with 54, A108. Transforms Ib, II, V.

64

CJB

Page 88: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 556

283 273

128145139 144

26 I44 41 1 24

46281=~ 4219

134 129 138

83 72

JDSII

Code: (283,273)(128,145)(139,144)(26,44,41, 17)(24, 138)(134,5) (129,46)(23,42)(28, 16) (12, 4)(8, 19)(83, 11)(72)

Elements: 4,5,8,11,12,16,17,19,23,24,26,28,41,42,44,46,72, 83,128,129,134,138,139,144,145,273,283

Remarks: In cluster with 70. Transform V.

65

Page 89: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 557

156

203 198

6096

49

78

154 59 44 42180

I 31 3328 I

46 ~ 85

40 52

200178 179

AJWD

Code: (203,156,198)(96,60)(18,180)(154,49)(78)(59,44, 42)(2, 33,85)(15,31)(46,28)(12,52)(40)(200)(1,179)(178)

Elements: 1,2,12,15,18,28,31,33,40,42,44,46,49,52,59,60,78, 85,96,154,156,178,179,180,198,200,203

66

Page 90: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 559

143161

255

125

179

57 66 5372

132

48 I 3919

7027 ~H31

51

219

172 168

AJWD

Code: (255,143,161)(125, 18)(179)(132,57,66)(53,72)(48,9)(39,70,19)(51,219)(27,14, 15,31) (13,1) (16) (172) (168)

Elements: 1,9,13,14,15,16,18,19,27,31,39,48,51,53,57,66,70, 72,125,132,143,161,168,172,179,219,255

67

Page 91: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 566

138

191

237

5385

3263

30

73 r--- 149

4350 117

134

47 ~22 3661

I

6481

226195

145

AJWD

Code: (237, 191, 138) (53, 85) (63,149,32) (117) (134, 73, 30) (13, 50) (43) (14, 36) (61,47,8)(22)(17,81,226)(64)(195)(145)

Elements: 8, 13, 14, 17,22,30,32,36,43,47,50,53,61,63,64, 73, 81, 85, 117, 134, 138,145,149,191,195,226,237

68

Page 92: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 574

170196 208

57

I 11339 120

56

59184 162

~454

11529 23

61

194 204176

JDSII

Code: (196,208,170) (57, 113) (184,12) (162,39,19) (20,56) (59) (54, 115) (35, 24)(17,7)(61)(29,6)(23)(10,204)(194)(176)

Elements: 6,7,10,12,17,19,20,23,24,29,35,39,54,56,57,59,61, 113,115,162,170,176,184,194,196,204,208

Remarks: In cluster with 74. Transform CJB-l.

69

Page 93: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 576

293 283

72 83

138139 144

42 ~ =t 28 4624

1 41 44I 26

145 154 149128

JDSII

Code: (293,283)(139,144) (138, 72, 83)(42,19,11)(8,12,28,46)(23, 4)(16)(44)(24,41) (26,154,5) (149) (145,17) (128)

Elements: 4,5,8,11,12,16,17,19,23,24,26,28,41,42,44,46,72, 83,128,138,139,144,145,149,154,283,293

Remarks: In cluster with 65. Transform V.

70

Page 94: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 585

297 288

122 132156

175

25 24 ~rl2764

97 3771

62141

113 29 101'--- 69 70

51 40

Code: (297,288)(156,132)(175,122)(97,25)(24,17,27,64)(7,10) (71,141)(37)(113,62)(101)(1,70)(29,69)(51,11)(40)

CJB

Elements: 1,7,10,11,17,24,25,27,29,37,40,51,62,64,69,70,71, 97,101,113,122,132,141,156,175,288,297

Remarks: In cluster with 83, A139. Transforms I, II, Va.

71

Page 95: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 587

275312

5984

34 1 25 132

i-----L

158 154 43 52 48

95

41 45 55 54 180

117

76 66 67 81

PJFITHW

Code: (312,275) (59, 84, 132) (158, 154) (34, 25) (9, 52,48) (43) (180) (95) (45, 55, 54)(117,41)(14,81)(76,10)(1,67)(66)

Elements: 1,9,10,14,25,34,41,43,45,48,52,54,55,59,66,67, 76,81,84,95,117,132,154,158,180,275,312

Remarks: In cluster with A162, A185. Transfonns I, Ib, IV.

72

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Order 27, reduced side 588

159

218 211

59

100

48

132 145 47 ~:- 64 16327

5740 51

29 I

8960

225 214

149

AJWD

Code: (218, 159,211) (59, 100) (48, 163) (132, 145) (47, 37, 64) (10, 27) (57) (40, 51)(89,29,14)(1,16,225)(15)(214)(60)(149)

Elements: 1, 10, 14,15,16,27,29,37,40,47,48,51,57,59,60,64,89, 100,132, 145, 149,159,163,211,214,218,225

73

Page 97: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 596

182 194220

57

I 11339 120

56

59206162

~454

11529 23

61

214196186

JDSII

Code: (220, ~82, 194) (57, 113, 12) (206) (162,39, 19) (20, 56) (59) (54, 115) (35, 24)(17,7)(61)(29,6)(23)(214)(10,196)(186)

Elements: 6,7,10,12,17,19,20,23,24,29,35,39,54,56,57,59,61, 113,115,162,182,186,194,196,206,214,220

Remarks: In cluster with 69. Transform CJB-1.

74

Page 98: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 600

262

338

112150

57 66105 110 38

39 ~7448 tJ:

54 49

188157 152

103

AJWD

Code: (338,262)(112,150)(105,57,66, 110)(74,38)(48,9)(39,20, 16) (188)(5,11)(19,1)(6)(17)(157,54)(49,152)(103)

Elements: 1,5,6,9,11,16,17,19,20,38,39,48,49,54,57,66,74, 103,105,110,112,150,152,157,188,262,338

75

Page 99: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 601

175 190

236

76 84

83 68 205153 92

70 8149 43

36 42

105

78212 206

96 87

JDSII

Code: (236,175,190)(76,84, 15)(205)(153,83)(68,8)(92)(70,81) (49,43)(212,11)(42,1)(206)(105,36)(78)(96,9)(87)

Elements: 1,8,9,11,15,36,42,43,49,68,70,76, 78,81,83,84,87,92,96,105,153,175,190,205,206,212,236

76

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Order 27, reduced side 605

148181

276

33

115

51 64 214

69 83 r------r-124 38 125

5255 41 IT56 63

36 I

205 210190

JDSII

Code: (276,148, 181)(115,33)(214)(51,64)(124,69,83)(38, 13)(25,52)(55,14)(63)(41,56)(36,16)(20,210)(205,15)(190)

Elements: 13,14,15, 16,20,25,33,36,38,41,51,52,55,56,63,64,69,83,115, 124,148,181,190,205,210,214,276

Remarks: In cluster with CM, C65, C82. Transforms Ib, II, V, V, CJB-8, CJB-lO.

77

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Order 27, reduced side 615

296319

78 129105 167136

~~ 4424 38

7188 91

53

160 52 49 152

103 99

51 50

JDSII

Code: (319,296)(129,167)(136,78,105)(34,44)(17, 88)(91, 38)(24,10)(71)(160)(53,152)(103,52,4)(49,99)(51,1)(50)

Elements: 1,4,10,17,24,34,38,44,49,50,51,52,53,71,78,88,91, 99,103,105,129,136,152,160,167,296,319

Remarks: In cluster with 93,97,98, A171, A175.

78

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Order 27, reduced side 620

155197

268

42 .

113

101 114 239

166

88 5275

~ 5136

I--

~41186 184163

87

AJWD

Code: (268,155,197)(113,42)(239)(101,114,166)(88, 13)(75,52) (186,22,8,2)(6,51,184)(14)(36)(163)(5,46)(41)(87)

Elements: 2,5,6,8,13,14,22,36,41,42,46,51,52,75,87,88,101, 113,114,155,163,166,184,186,197,239,268

Remarks: In cluster with C70. Transform I.

79

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Order 27, reduced side 620

281

339

126155

86125 128

29

39 I 47 97r---

33J:t-55 60

49 184156 22

11693

71

Code: (339,281) (126,155) (125, 86,128) (97, 29) (39, 47) (184) (156,8) (60,33,17,18)(55) (16, 1) (19) (49) (116) (93, 22) (71)

Elements: 1,8,16, 17,18,19,22,29,33,39,47,49,55,60,71,86,93, 97,116,125,126,128,155,156,184,281,339

Remarks: In cluster with A159. Transfonn I.

80

CJB

Page 104: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 621

162 177

282

147

192

55 42

h2971

114 60I 8752 37

1 4336

~88

56

252225

144

AJWD

Code: (282,162,177)(147,15)(192) (55, 42, 71,114)(87,60) (13, 29) (52,16)(36,37,43)(252)(88)(31,6)(25,225)(56)(144)

Elements: 6, 13, 15, 16,25,29,31,36,37,42,43,52,55,56,60, 71, 87, 88, 114, 144, 147,162,177,192,225,252,282

81

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Order 27, reduced side 627

283

344

128155

86123 135

6237 6649

77 7836 39 ~ 58

23

160 148112 111 96

Code: (344,283) (155,128) (123, 86,135) (66, 62) (37, 49) (77, 78) (160) (4,58) (39, 31)(148,36)(8,23)(112,1)(111,15)(96)

Elements: 1,4,8,15,23,31,36,37,39,49,58,62,66,77,78,86,96, 111,112,123,128,135,148,155,160,283,344

Remarks: In cluster with 94, A174. Transforms I, IlIa, V.

82

CJB

Page 106: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 631

288

343

51 l:- 132156

69 70113 29

62 27 ~~2471 64

97 37

25

175 187147

122

JDSII

Code: (343,288) (132,156) (113, 51, 40, 69, 70) (11, 29) (62) (97, 1) (71) (64,27,17,24)(10,7)(187)(37)(175)(25,147)(122)

Elements: 1,7,10,11,17,24,25,27,29,37,40,51,62,64,69,70,71, 97,113,122,132,147,156,175,187,288,343

Remarks: In cluster with 71, A139. Transforms I, II, Va.

83

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Order 27, reduced side 636

277

359

118159

56 6378

~41

162 J 7737 1 50

110

47 200164

11589

68

CJB

Code: (359,277)(118,159)(162,78,56,63)(77,41)(22, 14,20)(13, 50) (8,6) (2,37)(200)(110)(164)(115,47)(21,89)(68)

Elements: 2,6,8, 13, 14,20,21,22,37,41,47,50,56,63,68,77, 78, 89, 110, 115,118,159,162,164,200,277,359

Remarks: In cluster with A136. Transform I.

84

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Order 27, reduced side 640

289

351

129160

109 97

14531

55 42 9836

73 f--26

37 30 22r--'-

33 41 191

144107 124

74

Code: (351,289)(129,160)(145, 109,97)(98,31)(55,42)(191)(36,73) (16,26)(144,37)(30,22,3)(19)(124)(41)(107,33)(74)

Elements: 3, 16, 19,22,26,30,31,33,36,37,41,42,55,73,74,97,98, 107, 109, 124,129,144,145,160,191,289,351

Remarks: In cluster with A176. Transform I.

85

cm

Page 109: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 645

304341

77132

109 172

155

45 132

~3843

89

47 I 55

90 46

149169

102 105 120

JDSII

Code: (341,304) (132,172) (155,77,109) (45, 32)(89, 43) (13, 38, 90) (33,25) (3,169)(46)(8,55)(149,47)(15,120)(105)(102)

Elements: 3,8, 13, 15,25,32,33,38,43,45,46,47,55,77,89,90, 102, 105, 109, 120, 132,149,155,169,172,304,341

Remarks: In cluster with 92, A190, A214. Transforms I, Ib, II, III, IV, V.

86

Page 110: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 646

302344

65105

~132

112 94138 27 [23130

35 9026 '-- 59

69 52

212

164 149121

Code: (344,302) (105, 65,132) (138,112,94) (28, 37) (12, 16) (7, 30) (90,27) (23) (212)(35,59)(26,69,17)(52)(164)(149)(121)

Elements: 7,12,16,17,23,26,27,28,30,35,37,52,59,65,69,90,94, 105,112,121,132,138,149,164,212,302,344

Remarks: In cluster with A151. Transform 1.

87

CJB

Page 111: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 648

294

354

132162

89114

CJB

151

45'---

5329

69

50 ~

44 IW-~27

102

30

192

143

82111 120

Code: (354,294)(132, 162)(151, 114,89)(102,30)(50,39)(192)(45,69) (11,28)(44,17)(143,8)(27,18)(53)(120)(29,111)(82)

Elements: 8, 11, 17, 18,27, 28, ~9, 30, 39,44,45,50,53,69,82,89, 102,111,114,120,132,143,151,162,192,294,354

Remarks: In cluster with A196. Transform 1.

88

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Order 27, reduced side 650

161

253 236I

72 89

30 34 37 I

56 r~I

I105172 90

18499

I

95 10029

242213 195

JDSII

Code: (253,236,161)(72,89)(37,18,17)(30,172,34)(1, 105)(19) (184,56,13)(43)(90)(99)(5,100)(95)(29,242)(213)(195)

Elements: 1,5, 13, 17, 18, 19,29,30,34,37,43,56,72,89,90,95,99, 100,105,161,172,184,195,213,236,242,253

Remarks: In cluster with C79, C87. Transform Ib, II, V.

89

Page 113: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 656

181227

248

46

135

89

114134 184

95129

94 42 3874

H25 3432

245

200 211

AJWD

Code: (248,181,227)(135,46)(89,184)(114, 134)(129,95)(94,20)(74,42,38)(34,245)(13,25)(32,10)(1,12)(11)(211)(200)

Elements: 1,10,11,12,13,20,25,32,34,38,42,46,74,89,94,95, 114,129,134,135,181,184,200,211,227,245,248

Remarks: In cluster with 111, C75, C80. Transforms I, lb, II, III, IV, V.

90

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Order 27, reduced side 656

292

364

111

18171

133 41160 ~ 70

62 50

3968

47

~~37 30 183159

132 115

67

AJWD

Code: (364,292) (111,181) (160,133,71) (41, 70) (62, 9) (50) (68,183) (3,47) (39,159)(132,28)(16,23)(115)(37,7)(30)(67)

Elements: 3,7,9,16,23,28,30,37,39,41,47,50,62,67,68,70,71, 111,115,132,133,159,160,181,183,292,364

91

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Order 27, reduced side 660

304

356

132172

102 105149

43

47 I 55 89

90 46

~3845 132 184

155135

10977

Code: (356,304) (172, 132) (149,102,105) (43, 89) (47, 55) (90,15) (184, 3) (46) (155,33,8)(25,38)(135)(45,13)(32,109)(77)

Elements: 3,8, 13, 15,25,32,33,38,43,45,46,47,55,77,89,90, 102, 105,109, 132, 135,149,155,172,184,304,356

Remarks: In cluster with 86, A190, A214. Transforms I, Ib, II, III, IV, V.

92

CJB

Page 116: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 660

296

364

129

51 50167

103

160 52 49 38

9153

71 88

24

~144 197

136 144105

78

cm

Code: (364,296)(167,129)(160, 103,51,50)(1,49)(52)(38,91)(4, 197,53)(71,88)(144)(136,24)(10,44,17)(34)(105)(78)

Elements: 1,4, 10, 17,24,34,38,44,49,50,51,52,53, 71, 78,88,91, 103, 105, 129, 136,144,160,167,197,296,364

Remarks: In cluster with 78,97,98, A171, A175.

93

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Order 27, reduced side 663

319344

112 111 96

86

135 12323

37 36 39 ~ 5849

77 78

66 62

184 196155

128

JDSII

Code: (344,319)(112,111,96)(135,86, 123)(15,23,58)(49,37)(1,78,39,8)(36,77)(31)(196)(66,4) (62)(184)(155)(128)

Elements: 1,4,8,15,23,31,36,37,39,49,58,62,66,77,78,86,96, 111,112,123,128,135,155,184,196,319,344

Remarks: In cluster with 82, A174. Transforms I, IlIa, V.

94

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Order 27, reduced side 664

303

361

137166

96133 132

2937

59 10832 47R:- 53

"--------r;;- 39 195170

98 11685

Code: (361,303) (137,166) (133, 96,132) (108,29) (37, 59) (195) (32, 53, 47) (170) (50,9) (41) (39,8) (20, 33) (116) (98, 13) (85)

Elements: 8,9,13,20,29,32,33,37,39,41,47,50,53,59,85,96,98, 108,116,132,133,137,166,170,195,303,361

Remarks: In cluster with A223. Transfonn I.

95

CJB

Page 119: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 666

293

373·

114

179

83117 49 65

17334

82 6656

95 R;;-53 194

128120

67 81 76

JDSII

Code: (373,293) (114, 179) (173, 117,83) (49, 65) (34,82, 16) (66, 194) (95,56) (20,128)(120,53)(44,12)(32)(14,81)(76)(67)

Elements: 12,14, 16,20,32,34,44,49,53,56,65,66,67,76,81,82, 83,95,114,117,120,128,173,179,194,293,373

96

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Order 27, reduced side 668

296

372

129167

5378

105136 38

~~ 4424 144

71 88

205

160 52 49

103 99

51 50

JDSII

Code: (372,296)(129,167)(136, 78,105,53)(144,38)(34,44)(205)(17,88)(24,10)(71)(160)(103,52,4)(49,99)(51,1)(50)

Elements: 1,4,10,17,24,34,38,44,49,50,51,52,53,71,78,88,99, 103,105,129,136,144,160,167,205,296,372

Remarks: In cluster with 78,93,98, AI71, A175.

97

Page 121: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 671

319352

7850 51 105

13699

~4~ 44152 49 5224

71 8853

91

38

216

167 159129

JDSH

Code: (352,319) (136, 78,105) (152, 99, 50, 51) (34, 44) (49, 1) (52) (17, 88) (24,10)(71)(53,91,4)(216)(167,38)(159)(129)

Elements: 1,4,10,17,24,34,38,44,49,50,51,52,53,71,78,88,91, 99,105,129,136,152,159,167,216,319,352

Remarks: In cluster with 78,93,97, A171, A175.

98

Page 122: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 672

345 327

86116 147

143 180

129

57 I2846

71 544425 9336 f--

1--'- 3951

184165

27140

10578

AJWD

Code: (345,327)(180,147)(143,86, 116)(57,29)(28, 1)(71,46)(54,93)(184,44)(25,165,36)(15,39)(140)(51)(27,105)(78)

Elements: 1,15,25,27,28,29,36,39,44,46,51,54,57,71, 78,86,93, 105,116,140,143,147,165,180,184,327,345

99

Page 123: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 674

321353

87 133105

188161

69 4756 67 86 I-------T

39 1 3157 37 1 5346

160 165134

103 112

JDSII

Code: (353,321)(133, 188)(161, 105,87)(18,69)(86,47)(56,67)(39,8)(16,53)(31,165)(160,57)(46,37)(22,134)(112)(103)

Elements: 8,16,18,22,31,37,39,46,47,53,56,57,67,69,86,87, 103,105,112,133,134,160,161,165,188,321,353

Remarks: In cluster with 114, A212, A233. Transforms I, Ib, II, III, IV, V.

100

Page 124: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 676

169

263 244

94 75

83

103 108

165 44 l39 217

6241 36 29 I34

9256 63

248213 215

AJWD

Code: (263,169,244) (94, 75) (19, 83,217) (165,103, 108) (44,39) (62, 41) (5,34) (36,92,29)(63)(21,56)(248)(2,215)(213)

Elements: 2,5,19,21,29,34,36,39,41,44,56,62,63,75,83,92,94, ]03,108,165,169,213,215,217,244,248,263

101

Page 125: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 676

160

231

285

79 81

114 109 85 83

116 261 42169

11094119 68

53 6330

222242

212

JDSII

Code: (285,231,160)(79,81)(114, 109,8)(85,2)(83)(169, 116)(16,42,110)(5,94,26)(119)(68)(53,63)(30,242)(222)(212)

Elements: 2,5,8, 16,26,30,42,53,63,68,79,81,83,85,94, 109, 110, 114,116, 119, 160,169,212,222,231,242,285

Remarks: In cluster with 117, C78, C81, C95.

102

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Order 27, reduced side 676

292

384

123169

69 72108 46

135 77

66 47 ~

~37 r30 53

215184

157120

JDSII

Code: (384,292) (123, 169) (135, 69, 72, 108) (77, 46) (66,3) (47, 28) (215) (19, 9) (1,184)(10)(23,53)(157,37,7)(30)(120)

Elements: 1,3,7,9,10,19,23,28,30,37,46,47,53,66,69,72,77, 108,120,123,135,157,169,184,215,292,384

103

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Order 27, reduced side 677

206 194

277

tR 402871

95 137

24 97

143181 119

89

145

38 56

257219 201

AJWD

Code: (277,206,194)(17,40,137)(71,95,28,12)(7,10)(16,3) (13) (97)(181,143,24)(119)(89,145)(38,257,56)(219)(201)

Elements: 3,7, 10, 12, 13, 16, 17,24,28,38,40,56,71,89,95,97, 119, 137,143, 145, 181,194,201,206,219,257,277

104

Page 128: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 677

344 333

67 4956

h-;-80

118 137

170

128124

52 66 51

1-....-86

59 3545

163 149 140104 121

AJWD

Code: (344,333)(67,49,80,137)(170,118,56)(18,31)(128,13)(124) (52,66)(51,86)(163,59)(16,35)(45,149)(140)(121)(104)

Elements: 13, 16, 18,31,35,45,49,51,52,56,59,66,67,80,86, 104, 118, 121, 124,128,137,140,149,163,170,333,344

105

Page 129: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 678

312

366

5477

23

181

121 123153 100

5630

6524

-55 29 61

189 185159

71 74

AJWD

Code: (366,312)(54,77,181)(121,153, 123,23)(100)(65,56)(30, 189,4)(185)(24,159) (19, 61) (55,10) (29) (16, 13) (74) (71)

Elements: 4,10,13,16,19,23,24,29,30,54,55,56,61,65,71,74, 77, 100,121,123,153,159,181,185,189,312,366

106

Page 130: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 679

342 337

6897 105

94

145 261 42 170

30 -L-48 49 '-- 74

44 5891

50

192 174 172141

AJWD

Code: (342,337)(5,94,68,170)(145,97,105)(26,42)(30,74, 16)(48, 49)(91,14)(58)(44)(192,1)(50)(174,2)(172)(141)

Elements: 1,2,5,14,16,26,30,42,44,48,49,50,58,68,74,91,94, 97, ]05, 141,145,170,172,174,192,337,342

]07

Page 131: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 679

296

383

51 59I--

~36

231 186

81 94124 119 75

43 I 38 2569 64

177197

172133

CJB

Code: (383,296)(51,59,186)(36,15)(7,52)(22)(124,81, 119,94, 1) (23)(75)(43,38)(25,69)(64,197)(5,177)(172)(133)

Elements: 1,5,7,15,22,23,25,36,38,43,51,52,59,64,69,75,81, 94,119,124,133,172,177,186,197,296,383

Remarks: In cluster with SPSR 25:296x592. Transform VIII.

108

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Order 27, reduced side 680

177

239264

85 92

114 125 cl~ 738947

57175

86106 12095 98

241221 218

AJWD

Code: (264,239,177)(85,92)(114,125)(45,33,7) (175, 89)(26, 73) (12,47)(57)(120)(86,106,11)(95,98)(241,20)(221)(218)

Elements: 7,11,12,20,26,33,45,47,57,73,85,86,89,92,95,98, 106,114,120,125,175,177,218,221,239,241,264

109

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Order 27, reduced side 680

312

368

61 64

52 47 25 53 187

157 159

107 70

7667 48

181155 153

86 105

AJWD

Code: (368,312)(61,64,187)(157,159,52)(47, 14)(11,53)(25)(8,17) (107)(70)(76,181) (155, 2) (153, 67, 48) (19, 105) (86)

Elements: 2,8,11,14,17,19,25,47,48,52,53,61,64,67,70,76,86, 105,107,153,155,157,159,181,187,312,368

110

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Order 27, reduced side 690

181227

282

46

13589

34184

114134

95

163

94 42 3874

32 H25

279

200 211

AJWD

Code: (282,181,227) (135,46) (89,184) (114,134,34) (163, 95) (94, 20) (279) (74,42,38)(13,25) (32, 10)(1, 12)(11) (211)(200)

Elements: 1,10,11,12,13,20,25,32,34,38,42,46,74,89,94,95, 114,134,135,163,181,184,200,211,227,279,282

Remarks: In cluster with 90, C75, C80. Transforms I, Ib, II, III, IV, V.

111

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Order 27, reduced side 692

300

392

127173

69 71

136116 46

~4081

67261 I

39 45 41

219

164184

125

AJWD

Code: (392,300)(127,173)(136,69,71, 116)(81,46)(67,2)(33,40) (219)(26,7)(19,23,5)(18,184)(45)(41)(164,39)(125)

Elements: 2,5,7,18,19,23,26,33,39,40,41,45,46,67,69,71,81, 116,125,127,136,164,173,184,219,300,392

112

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Order 27, reduced side 693 AJWD

349 344

165189

154165

5266

40 3111-24--'-r- --.---'---i f----l.-....-------i

50 83

91 ~44 38

155 164

88 85 89112

Code: (349,344) (5, 154, 185) (165, 189){40, 83, 31) (91, 50, 24) (52, 164) (32,155,66)(44,6)(38)(23,112)(89)(88,3)(85)

Elements: 3,5,6,23,24,31,32,38,40,44,50,52,66,83,85,88,89, 91,112,154,155,164,165,185,189,344,349

113

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Order 27, reduced side 696

321

375

133

103188

112160

47

57 46 371 53

h-- 86

31 I 39

56 6769

187161 156

105 87

Code: (375,321)(188, 133)(160, 103,112)(47,86)(57,46)(37,53,22) (8,39)(187,31)(67,16)(161,56)(69)(156)(105,18)(87)

Elements: 8, 16, 18,22,31,37,39,46,47,53,56,57,67,69,86,87, 103,105, 112,133,156,160,161,187,188,321,375

Remarks: In cluster with 100, A212, A233. Transforms I, Ib, II, III, IV, V.

114

CJB

Page 138: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 703

307

396

119

188

5828L

........... 69174 164 32

76

30 8146

41 51208

133 143 12792

AJWD

Code: (396,307)(119,188)(174,164,58)(28,22,69)(7, 15)(76,9, 1) (8)(32)(81,208)(30,46)(51,143)(133,41)(127)(92)

Elements: 1,7,8,9,15,22,28,30,32,41,46,51,58,69,76,81,92, 119,127,133,143,164,174,188,208,307,396

115

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Order 27, reduced side 704

323

381

69 75

74 63179

90 88129 81

5545 41 825892 27

49

194 202167

141

JDSII

Code: (381,323) (69, 75,179) (90,88,129,74) (63, 6) (81) (55, 82) (2, 45,41) (92) (58,23) (202) (4,194,27) (49) (167) (141)

Elements: 2,4,6,23,27,41,45,49,55,58,63,69,74, 75,81,82,88,90,92,129,141,167,179,194,202,323,381

116

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Order 27, reduced side 706

160

231

315

79 81

114 109 85 83

30261 42

116 11094169 68

149

53 63

272

222 212

JDSII

Code: (315,231,160) (79,81) (114, 109, 8) (85, 2) (83) (169, 116,30) (16,42, 110) (5,94,26)(149)(68)(53,63)(272)(222)(212)

Elements: 2,5,8, 16,26,30,42,53,63,68,79,81,83,85,94, 109, 110, 114, 116, 149, 160,169,212,222,231,272,315

Remarks: In cluster with 102, cn, C81, C95.

117

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Order 27, reduced side 708

179

235

294

56

123

67115 109

48111

4465175 150 61 -r-

102

64 4784 86

43

239 256213

AJWD

Code: (294,179,235)(123,56)(67,115, 109)(175, 111,8)(150,48)(44,65)(102,61)(23,21)(64,47)(86)(84)(256,4~)(239)(213)

Elements: 8,21,23,43,44,47,48,56,61,64,65,67,84,86, 102, 109, 111, 115, 123, 150,175,179,213,235,239,256,294

118

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Order 27, reduced side 708

177

228

303

51

126

99

24 18076 66

94 4281

16937 49 I 59

39

--75

56

10864

300

236

172

AJWD

Code: (303,177,228) (126, 51) (99,180) (169,94,76,66,24) (42, 81) (l0, 59, 39) (37,49)(75,19)(300)(56)(108)(236,64)(172)

Elements: 10,19,24,37,39,42,49,51,56,59,64,66,75,76,81,94, 99,108,126,169,172,177,180,228,236,300,303

119

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Order 27, reduced side 708

342366

4431 I

~ 53132

138166

177

95 34

51 98

'--- 87 385763

27 200165

117 13690

AJWD

Code: (366,342) (44,132, 166) (177,138,31,20) (11, 53) (42) (95) (98, 34) (51,87) (200)(57,38) (165,12) (63) (136) (27,117) (90)

Elements: 11, 12,20,27,31,34,38,42,44,51,53,57,63,87,90,95, 98,117,132,136,138,165,166,177,200,342,366

120

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Order 27, reduced side 709

299

410

61 73f--

49 165

8548

142 12251 I147 4475

128

95

157 177152

106 117

AJWD

Code: (410,299)(61,73,165)(49,12)(85)(1,48)(147, 142, 122)(75,51,7)(44,128)(95)(20,177)(5,157)(152)(11,117)(106)

Elements: 1,5,7,11,12,20,44,48,49,51,61,73,75,85,95,106, 117,122,128,142,147,152,157,165,177,299,410

121

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Order 27, reduced side 716

199240

277

77122 36

98 11354 204

17968

81 57 1 4359 ~8 55

100 84

260 272

184

JDSH

Code: (277,199,240) (122, 77) (36, 204) (113) (179, 98) (54, 68) (81, 57, 14) (43, 59, 38,55)(21,17)(100)(4,272)(84) (260)(184)

Elements: 4,14,17,21,36,38,43,54,55,57,59,68,77,81,84,98, 100,113,122,179,184,199,204,240,260,272,277

122

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Order 27, reduced side 716

343373

138113 108

152 205

3974 79 76 91

37

58 1 43 44 ~~191

168156

101 100

AJWD

Code: (373,343) (138, 205) (152, 113, 108) (79, 76, 91) (39, 74) (191) (37, 168) (3, 58,15) (156) (43, 44,19) (6, 31) (25) (101) (100)

Elements: 3,6,15,19,25,31,37,39,43,44,58,74,76,79,91,100, 101,108,113,138,152,156,168,191,205,343,373

123

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Order 27, reduced side 724

365 359

43 49 I 63 7682 99147 78 165

64

65 89

9472

9254

212 200166 146

AJWD

Code: (365,359) (6, 49, 63, 76,165) (147, 82, 99, 43) (78, 14) (64,13) (89) (65,17) (94,22)(72,92)(212)(54,200)(166)(146)

Elements: 6,13,14,17,22,43,49,54,63,64,65,72,76, 78,82,89,92,94,99,146,147,165,166,200,212,359,365

124

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Order 27, reduced side 728

213265 250

I

108 105

56130 120

209 80 87 102

40

100 11073

94 8260

28

254 248226

AJWD

Code: (265,213,250) (108,105) (130,120) (209, 56) (3, 102) (80,87) (10, 110) (40,100)(73,7)(94)(82,60)(254,28)(22,248)(226)

Elements: 3, 7, 10,22,28,40,56,60, 73, 80, 82, 87, 94, 100, 102, 105, 108, 110, 120, 130,209,213,226,248,250,254,265

125

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Order 27, reduced side 731

224 227

280

67 54 51103

89 17678

42 I105

61180

I

91 99 8580 65

131

66

271 263

197

JDSII

Code: (280,224,227) (67,103,54) (51,176) (105) (l80, 89,11) (78) (42, 61) (91, 99,19)(85,20)(80)(65,131)(271)(263,66)(197)

Elements: 11,19,20,42,51,54,61,65,66,67,78,80,85,89,91,99, 103,105,131,176,180,197,224,227,263,271,280

126

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Order 27, reduced side 732

206 212

314

200218

129

18546

56 I 64 r:- 130106

136

3988 - 30

49

233196

166137

AJWD

Code: (314,206,212) (200,6) (218) (185, 129) (46, 130,24) (106, 136) (56,64,9) (55)(233,8)(88,39)(10,196,30)(49)(166)(137)

Elements: 6,8,9,10,24,30,39,46,49,55,56,64,88,106,129,130, ]36, ]37, 166, 185,196,200,206,212,218,233,314

127

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Order 27, reduced side 732

206 212

314

200218

113 t

201

2546

88 13010671 136

64 I 56 ~3 30

217196

153 166

JDSII

Code: (314,206,212) (200, 6) (218) (201,113) (46,130,24) (88, 25) (106, 136) (71) (217,64,8)(56,23)(10,196,30)(33)(166)(153)

Elements: 6,8, 10,23,24,25,30,33,46,56,64, 71, 88, 106, 113, 130, 136, 153, 166, 196,200,201,206,212,217,218,314

128

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Order 27, reduced side 733

324

409

81 63

~

3~ 5327 45 180

9041

72157 147 46

57 74 69

229

167 177 160

AJWD

Code: (409,324)(81,63,180)(18,45)(19,53,27)(157, 147,90, 15)(34) (72)(41,46)(57,74)(69,229)(10,177,17)(167)(160)

Elements: 10,15,17,18,19,27,34,41,45,46,53,57,63,69,72, 74,81,90,147,157,160,167,177,180,229,324,409

129

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Order 27, reduced side 736

191212

333

5332

138

85180

63

152 156185

117 126

33

49 4092 86

43 46

251218

135 132

AJWD

Code: (333,191,212)(138,53)(32,180)(85)(4, 156,63)(185, 152)(117,126)(33,251,24)(92,49)(218)(40,86)(43,46)(135)(132)

Elements: 4,24,32,33,40,43,46,49,53,63,85,86,92,117,126, 132, 135,138, 152, 156,180,185,191,212,218,251,333

130

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Order 27, reduced side 744

222272

250

102 12052 89

161

1-'--66220 43136 118

55 I 48 I 42

32 12583

284252

208

AJWD

Code: (272,222,250)(102,120)(89,161)(220,52)(136,18)(23,66) (118,20)(43)(55,48,6)(42,125)(7,83)(32,284)(252)(208)

Elements: 6,7,18,20,23,32,42,43,48,52,55,66,83,89,102,118, 120,125,136,161,208,220,222,250,252,272,284

131

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Order 27, reduced side 744

222272

250

102 12052 101

149

220 136rs11 42 I 48

11867 66

32 137

71

284252

208

AJWD

Code: (272,222,250)(102,120)(101,149)(220,52)(136,18)(118,20) (11,42,48) (31)(67,6)(66,137)(32,284,5)(71)(252)(208)

Elements: 5,6, 11, 18,20,31,32,42,48,52,66,67, 71, 101, 102, 118, 120, 136, 137, 149,208,220,222,250,252,272,284

132

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Order 27, reduced side 752

210251

291

41 I 47~

12253

102 94 90

62 ~

22998 126 135

164

74

285

232 235

JDSII

Code: (291,251,210) (41, 47, 122) (102, 94, 90, 6) (53) (229, 62) (31,22) (9, 135) (4,126)(98)(164)(74,24)(285)(3,235)(232)

Elements: 3,4,6,9,22,24,31,41,47,53,62,74,90,94,98,102,122, 126,135,164,210,229,232,235,251,285,291

133

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Order 27, reduced side 752

208245

299

52 56100

4879

46111 102 - 226

57 68188

77 65 60

136

70

265 281

206

AJWD

Code: (299,208,245)(100,52,56) (19, 226)(48,4) (79)(188,111) (102,46)(11,68)(57)(77,136)(65,60)(5,281)(70)(265)(206)

Elements:. 4,5,11, 19,46,48,52,56,57,60,65,68,70,77,79,100, 102,111, 136, 188,206,208,226,245,265,281,299

134

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Order 27, reduced side 755

216

276 263

60 47

109

57 71 93I-- 31

43 29 1 78 217208 271 5645

1--...... 124

63143

66

271 275

209

AJWD

Code: (276,216,263)(60,109,47)(93,217)(208,57,71)(78,31) (43, 14)(29,56) (124)(45,27)(18,143)(63)(66,275)(271)(209)

Elements: 14,18,27,29,31,43,45,47,56,57,60,63,66,71,78,93, 109,124,143,208,209,216,217,263,271,275,276

135

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Order 27, reduced side 755

215

278 262

88127 41

64

129 221214 101 90

49 36 rd42 51103

56 61

263 272220

JDSII

Code: (278,215,262) (127, 88) (41, 221) (214, 64) (129) (101, 90) (23, 42, 103,51) (49,36,16)(4,19)(20)(61)(56)(272)(263)(220)

Elements: 4,16,19,20,23,36,41,42,49,51,56,61,64,88,90,101, 103,127,129,214,215,220,221,262,263,272,278

136

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Order 27, reduced side 755

189

262304

78111

37 41 38

56 5398

141 224183 55 54

8545

12296

268 269218

AJWD

Code: (304, 189,262) (78, 111) (38, 224) (37, 41) (8, 141) (183, 98, 56,4) (53) (55, 1)(54)(85,13)(122)(96,45)(269)(268)(218)

Elements: 1,4,8,13,37,38,41,45,53,54,55,56,78,85,96,98,111, 122,141,183,189,218,224,262,268,269,304

137

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Order 27, reduced side 764

199240

325

4167

91r-----'"- 50

43r-- 4627

119 23173

148193

62

45 l~130

6629

293246

225

JDSII

Code: (325, 199,240) (67,91,41) (50, 231) (43, 24) (46, 119) (16, 27) (193, 148) (73)(130,62)(293)(45,37,66)(8,29)(246)(225)

Elements: 8,16,24,27,29,37,41,43,45,46,50,62,66,67,73,91, 119,130,148,193,199,225,231,240,246,293,325

138

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Order 27, reduced side 807

216259

332

3871

107 33 28

13298 236

137

20448 5452 76

67114 110

312271

224

AJWD

Code: (332,216,259)(107, 71,38)(33,5)(28,236)(132)(9,98)(204,137)(2,54,76)(48,52)(67,114,4)(110)(312)(271)(224)

Elements: 2,4,5,9,28,33,38,48,52,54,67,71, 76, 98, 107, 110, 114, 132, 137, 204,216,224,236,259,271,312,332

139

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Order 27, reduced side 832

388444

190 198114 131

199

97 80 68 65

12587

111

6083

6333

189177129 135

106 96

Code: (444,388)(190,198)(114,131, 199)(97,17)(80,68)(65, 125)(87,111)(189,83,60) (177) (63, 24) (135) (23,129,33)(106)(96)

CJB

Elements: 17,23,24,33,60,63,65,68,80,83,87,96,97,106,111, 114,125,129,131,135,177,189,190,198,199,388,444

Remarks: In cluster with 158, A256. Transforms I, lIe, Va.

140

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Order 27, reduced side 835

228287

320

59

169

4776

I29h 161185

197

121171

96 113137

318267 250

JDSII

Code: (320,228,287)(169,59)(185,161)(197,76,47)(29, 18)(11,7) (4,3)(1,2)(121)(171)(24,137)(96,113)(318)(267)(250)

Elements: 1,2,3,4,7,11,18,24,29,47,59,76,96,113,121,137, 161,169,171,185,197,228,250,267,287,318,320

141

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Order 27, reduced side 844

392

452

181211

207245

30

151

3868

101 I 51 58 241I 3150

'-- 147120

84 96 8967

Code: (452,392)(181,211)(207,245)(151,30)(241)(101,68,38)(27, 51,147,58)(3,24)(31,120)(50,21)(96)(89)(84,17)(67)

Elements: 3,17,21,24,27,30,31,38,50,51,58,67,68,84,89,96, 101,120,147,151,181,207,211,241,245,392,452

Remarks: In cluster with A341, A373. Transforms I, Ib, IV.

142

CJB

Page 166: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 855

432 423

108

171 153 184

63 1 45239

75105 83

~101

46

N4~252193

165 14798

Code: (432,423) (184, 239) (171,108,153)(63,45)(18,75,105)(252) (83,101)(43,32)(46,193)(23,165)(11,21)(54)(147)(44)(98)

Elements: 11, 18,21,23,32,43,44,45,46,54,63,75,83,98, 101, 106, 108,147, 153, 165,171,184,193,239,252,423,432

Remarks: In cluster with A288. Transform 1.

143

CJB

Page 167: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 855

404451

101119

184129 151171 83

42 ~19 48137

r---- 5867

93

267233

195160

CJB

Code: (451,404)(119,101,184)(171, 129, 151)(18,83)(137)(42,22, 17,48)(267)(5,12)(20,7)(93,58)(19)(233)(67)(195)(160)

Elements: 5,7,12,17,18,19,20,22,42,48,58,67,83,93,101,119, 129,137,151,160.171,184,195,233,267,404,451

Remarks: In cluster with A262. Transform I.

144

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Order 27, reduced side 860

383

477

168215

128160

189 47

12161 67

75 8556

~ 262

37141 194 188138

78

Code: (477,383) (168, 215) (160, 189, 128) (121, 47) (262) (61, 67) (75, 85) (56, 194)(188) (33, 29,13) (3, 138) (16) (4, 41) (37) (78)

Elements: 3,4,13,16,29,33,37,41,47,56,61,67,75,78,85,121, 128,138,160,168,188,189,194,215,262,383,477

Remarks: In cluster with A270. Transform 1.

145

CJB

Page 169: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 872

363

509

165198

1122 W- 108

200 187 75 53127

65 57

50 I 63236

163 176 184

113

Code: (509,363) (198, 165) (200, 187, 122) (29, 28, 108) (1, 27) (4,26) (127,75) (53)(65,57)(236)(184)(13,63,176)(163,50)(113)

Elements: 1,4, 13,26,27,28,29,50,53,57,63,65, 75, 108, 113, 122, 127, 163, 165, 176,184,187,198,200,236,363,509

Remarks: In cluster with A284. Transform I.

146

CJB

Page 170: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 876

404

472

213 191

145 144183

38 63 81 80107 f-- 169

45 155

221

152179 51 57

104 11253 55

PJFITHW

Code: (472,404)(213,191)(183,145,144)(22, 169)(63,81)(38, 107) (80,155)(221)(45,18)(179)(152)(57,112)(104,51)(2,55)(53)

Elements: 2,18,22,38,45,51,53,55,57,63,80,81,104,107,112, 144,145,152,155,169,179,183,191,213,221,404,472

Remarks: In cluster with A300. Transform 1.

147

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Order 27, reduced side 885

405

480

102132

171

120133

69 I175 36

185129

55 6589 97 90

111 105

230 220 219

109 107

CJB

Code: (480,405)(171,102,132)(175,120,185)(69, 33)(3,129)(36)(89,97,90)(55,65)(230)(220,111,8)(219)(105)(109,2)(107)

Elements: 2,3,8,33,36,55,65,69,89,90,97, 102, 105, 107, 109, 111, 120, 129, 132, 171,175,185,219,220,230,405,480

Remarks: In cluster with 160, A266. Transforms II, V, CJB-28.

148

Page 172: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 892

400

492

177223

143 16318646

43131

100 5357 49 64 87

82 g269

157184

132 150

cm

Code: (492,400) (177,223) (143,186,163)(131,46)(269)(100,43)(23, 87,53)(57,82,49,64)(184)(33,16)(157)(1,150)(17)(132)

Elements: 1,16,17,23,33,43,46,49,53,57,64,82,87,100,131, 132,143,150,157,163,177,184,186,223,269,400,492

Remarks: In cluster with A319. Transform 1.

149

Page 173: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 895

388

507

100 123165

77

139 ~ 146189 179 93

57 92

40

249 238199 209

CJB

Code: (507,388) (165, 100, 123) (77, 23) (189, 179,139) (146) (93, 36, 15, 9, 12) (6, 3)(92)(21)(57)(40,249)(238)(10,209)(199)

Elements: 3,6,9,10,12,15,21,23,36,40,57,77,92,93,100,123, 139,146,165,179,189,199,209,238,249,388,507

Remarks: In cluster with A259. Transform II.

150

Page 174: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 897

396

501

6899

t--.---37 31

22973

113 130165 '----r---

187 33 40........f-l- 5236

98 87

272

209 231185

CJB

Code: (501,396) (68, 99,229) (37,31) (130) (187, 165,73, 113) (33, 40) (23, 10) (3,52,98)(13)(87,272)(36)(22,231)(209)(185)

Elements: 3,10,13,22,23,31,33,36,37,40,52,68,73,87,98,99, 113,130,165,185,187,209,229,231,272,396,501

Remarks: In cluster with SPSR 25:396x792. Transform VIII.

151

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Order 27, reduced side 900

407

493

78122

70 207

128 37

178 ~~ 85195

50 ~H 46113

60

286

212 229173

TRW

Code: (493,407)(78,122,207)(8,70)(195,178, 128)(37,85)(59, 11) (48)(113,286)(50,32,46)(18,14)(60)(17,229)(212)(173)

Elements: 8,11,14,17,18,32,37,46,48,50,59,60,70,78,85,113, 122,128,173,178,195,207,212,229,286,407,493

Remarks: In cluster with SPSR 25:407x814. Transform VIII.

152

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Order 27, reduced side 900

404

496

199 205

124176 196

50 4452 72

149 1619460

~~ 54228 20861 31

148 13087 99

Code: (496,404) (199,205) (176,124,196) (149, 50) (44,161) (52, 72) (94) (228) (208,60)(40,54)(148,61)(31,130)(26,14)(99)(87)

Elements: 14,26,31,40,44,50,52,54,60,61,72,87,94,99,124,130, 148,149,161,176,196,199,205,208,228,404,496

Remarks: In cluster with 177, A370. Transforms I, IIIa, V.

153

CJB

Page 177: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 900

401

499

176225

90120 136153

63 L 49

H: 12799 83 69

248274

182 196

Code: (499,401) (176, 225) (153, 90,120,136) (127, 49) (63, 27) (19,8) (5,99,16) (13)(274)(17,2)(83,69)(15)(248)(196)(182)

Elements: 2,5,8,13,15,16,17,19,27,49,63,69,83,90,99,120,127, 136,153,176,182,196,225,248,274,401,499

Remarks: In cluster with A312. Transform 1.

154

CJB

Page 178: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 905

431474

130 137164

231 243123 116 f--~

59 133

33 4431

~ol 69

200 221188 177

119

Code: (474,431)(130,137,164)(231,243)(123, 7)(116,28)(1,30,133) (29)(59)(33,221,44)(200,31)(19,69,188)(177)(50)(119)

cm

Elements: 1,7,19,28,29,30,31,33,44,50,59,69,116,119,123, 130,133,137,164,177,188,200,221,231,243,431,474

Remarks: In cluster with 167, 194, A277, A346.

155

Page 179: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 906

414

492

145 136 133

129 31 34181 182 f---

96 9965

52 15477

39

119101

233260

193

107 113

PJPITHW

Code: (492,414) (145, 136, 133) (181, 129, 182) (34, 99) (9, 96, 31) (154) (65) (52, 77)(260)(233)(24,119,39)(101)(193)(6,113)(107)

Elements: 6,9,24,31,34,39,52,65,77,96,99,101,107,113,119,129, 133,136,145,154,181,182,193,233,260,414,492

Remarks: In cluster with A324. Transform 1.

156

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Order 27, reduced side 908

403

505

176227

120161

224 51

41 12579

108 94

4533 278~ 61

204179

~~134 r-- 37

63 50

Code: (505,403) (176, 227)(224,161,120)(125,51)(41,79)(278) (108, 94)(204)(179,45)(33,61)(134,19)(52)(24,37)(63,13)(50)

Elements: 13, 19,24,33,37,41,45,50,51,52,61,63, 79, 94, 108, 120, 125, 134, 161, 176,179,204,224,227,278,403,505

Remarks: In cluster with A311. Transform I.

157

CJB

Page 181: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 910

466 444

96114 131106

135 129199

3363

83 97 80 6860

111 87 12565

255 267

198 190

Code: (466,444) (114,131,199) (135, 96,129,106) (97,17) (63, 33) (23, 83) (80, 68)(125,60) (111, 24) (87) (267) (65, 255) (198)(190)

Elements: 17,23,24,33,60,63,65,68,80,83,87,96,97,106,111, 114,125,129,131,135,190,198,199,255,267,444,466

Remarks: In cluster with 140, A256. Transforms I, lIe, Va.

158

CJB

Page 182: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 913

260296

357

36

224

127

230 332

160191

57

47173

116 101

28576 75 86 238

153

77 78 82

AJWD

Code: (357,260,296)(224,36)(332)(230,127)(160,191)(173, 57)(47, 285)(116,101)(238)(15,86)(153,76,75)(4,82)(1,78)(77)

Elements: 1,4,15,36,47,57,75,76,77,78,82,86,101,116,127, 153, 160,173,191,224,230,238,260,285,296,332,357

159

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Order 27, reduced side 915

435480

109 107

120 219185 175

111 105

65 55

89 97 90129

136250 260 69 I33

171132

102

JDSII

Code: (480,435)(109;107,219)(185,120, 175)(2, 105)(111)(65,55)(8,97)(90,129)(260,89)(250)(171,69,36)(33,3)(132)(102)

Elements: 2,3,8,33,36,55,65,69,89,90,97,102,105,107,109,111, 120,129,132,171,175,185,219,250,260,435,480

Remarks: In cluster with 148, A266. Transforms II, V, CJB-28.

160

Page 184: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 920

407

513

177230

85 107139

182 I 5363 1 41 124

4368 ~O~ 59

88

283225 200 212

Code: (513,407)(177,230)(182, 139,85, 107)(124,53)(63,22)(41,88) (283)(43,68,28)(19,26,59)(40,7)(33)(225)(212)(200)

Elements: 7,19,22,26,28,33,40,41,43,53,59,63,68,85,88,107, 124,139,177,182,200,21~,225,230,283,407,513

Remarks: In cluster with A318. Transform 1.

]61

CJB

Page 185: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 920

396

524

176220

79129 146 43

170

~~ 13389

35 131112

71 4299

250215

155 173127

AJWD

Code: (524,396)(176,220)(170,129,146,79)(43,133)(67, 12)(55)(89, 131)(112,17)(250,35)(99,71)(215,42)(173)(28,155)(127)

Elements: 12,17,28,35,42,43,55,67,71,79,89,99,112,127,129, 131,133,146,155,170,173,176,215,220,250,396,524

162

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Order 27, reduced side 924

267 284

373

250

146 301

227

81 65 67

183

116 105

196

85 95324

224 45 49

139 14394

AJWD

Code: (373,267,284)(250,17)(301)(227, 146)(67, 183)(81,65)(16, 116)(105,196)(324)(10,95)(224,85)(4,49,143)(139,45)(94)

Elements: 4,10,16,17,45,49,65,67,81,85,94,95,105,116,139, 143,146,183,196,224,227,250,267,284,301,324,373

163

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Order 27, reduced side 927

419

508

182237

168 148192

55

127

9772

91 7780

112

321 61 78271

I 174199

11586 82

CJB

Code: (508,419)(237,182)(192, 168,148)(55,127)(97,271,72)(91,77) (112,80)(199)(174)(13,78)(32,61)(115,29)(86,4)(82)

Elements: 4,13,29,32,55,61,72,77,78,80,82,86,91,97,112,115, 127,148,168,174,182,192,199,237,271,419,508

Remarks: In cluster with A347. Transform II.

164

Page 188: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 928

395

533

163

232

140 144 69119

224

~8436 32

n 160 68 301

251

171

94 111

JDSII

Code: (533,395) (163, 232) (224,140,144,25) (119, 69) (301) (84, 40,16) (12,251) (28)(36,4)(32)(68) (171, 77, 60) (17, 111)(94)

Elements: 4,12,16,17,25,28,32,36,40,60,68,69,77,84,94,111, 119,140,144,163,171,224,232,251,301,395,533

165

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Order 27, reduced side 932

449483

3469 J 75

132 13963

81

39257 260

83 1;- 100171

65 62

244192 189 180

127

AJWD

Code: (483,449) (34,132,69,75,139) (257, Z60) (63, 6) (81) (171, 24) (39,100) (83, 22)(61)(244)(192,65)(62,189,9)(180)(127)

Elements: 6,9,22,24,34,39,61,62,63;65,69,75,81,83,100,127, 132,139,171,180,189,192,244,257,260,449,483

166

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Order 27, reduced side 938

431

507

137 130164

119

200 188-'-- 116 123

~O~ 69 133 59

33 44

231 243 254210

Code: (507,431)(164,137,130)(200,119,188)(7, 123)(28, 116)(133,30,1)(29)(59)(50,69)(31,19)(44,254)(243,33)(231)(210)

Elements: 1,7, 19,28,29,30,31,33,44,50,59,69, 116, 119,123, 130, 133, 137, 164, 188,200,210,231,243,254,431,507

Remarks: In cluster with 155, 194, A277, A346.

167

CJB

Page 191: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 939

428

511

83 107

238

159 170 131241

5996g 100 89 81

54 43 136 273

187 169 177133

CJB

Code: (511,428) (83,107,238) (241,159,170,24) (131) (96, 273) (59, 100) (89,81) (34,25)(187,54)(9,16)(177)(43)(36,169)(133)

Elements: 9,16,24,25,34,36,43,54,59,81,83,89,96,100,107, 131,133,159,169,170,177,187,238,241,273,428,511

Remarks: In cluster with SPSR 25:428x856. Transform VIII.

168

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Order 27, reduced side 945

446499

194146 153 252

200

54162~9 63131 r--

14468

109

247246

149 174129

Code: (499,446) (194,252) (200, 146, 153) (131,63) (54, 62, 23, 7) (16, 144) (39) (5,247)(246,8)(68)(109)(25,174)(20,149)(129)

Elements: 5,7,8,16,20,23,25,39,54,62,63,68, 109, 129, 131, 144, 146,149,153,174,194,200,246,247,252,446,499

Remarks: In cluster with 185, A400, A427. Transforms I, lb, II, III, IV, V.

169

CJB

Page 193: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 945

440

505

65110

45

151 265197 177

155

46105

79 98 75~36

243'44l 69

240184 165

113

Code: (505,440)(65, 110,265)(197,151,177,45)(155)(46,105) (79, 98) (243)(75,47,58,240)(36,11)(184)(69)(165,8)(44)(113)

Elements: 8,11,36,44,45,46,47,58,65,69,75,79,98,105,110,113, 151,155,165,177,184,197,240,243,265,440,505

Remarks: In cluster with SPSR 25:440x880. Transform VIII.

170

CJB

Page 194: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 948

486 462

134 129 153

247 285

113f-4~ 73

183

1205590 t-- 84

35 33215 201

125 140 150117

AJWD

Code: (486,462) (24,153,285) (247,134,129) (26, 73,183) (113, 21) (47) (120) (215,90,55)(84,201)(35,20)(150,33)(140)(125)(117)

Elements: 20,21,24,26,33,35,47,55,73,84,90,113,117,120, 125, 129,134,140,150,153,183,201,215,247,285,462,486

171

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Order 27, reduced side 953

457496

7334

180 204

194 107

268

147 140 8072 124

122f---

f--- 36 4450

79

189 169 176 168141

110

AJWD

Code: (496,457)(73,180,204)(268,194,34) (107)(147, 140)(80,124) (122,72)(36,44)(50,22)(189,79)(176)(169)(168)(31,141)(110)

Elements: 22,31,34,36,44,50,72,73,79,80,107,110,122,124, 140,141,147,168,169,176,180,189,194,204,268,457,496

172

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Order 27, reduced side 955

439

516

I

141 150 148

115

191 4868 I210 93 66 8084

42 4576

104 92

282229 248

196

AJWD

Code: (516,439)(141,150,148)(210, 191, 115)(48,93)(68,80)(84,66) (3,45)(76,42)(18,104,12)(92)(282)(19,248)(229)(196)

Elements: 3, 12, 18, 19,42,45,48,66,68, 76, 80, 84, 92,93, 104, 115, 141, 148, 150, 191,196,210,229,248,282,439,516

173

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Order 27, reduced side 955

422

533

77125

134143 22042r 35

127 6795

65183 198

56 71 134

45

313

239 224179

CJB

Code: (533,422) (77,125,220) (34, 43) (183,127,198,42,17) (8, 35) (30,95) (25) (67)(65)(134,313)(56,71)(239)(224,45)(179)

Elements: 8,17,25,30,34,35,42,43,45,56,65,67,71,77,95,125, 127,134,179,183,198,220,224,239,313,422,533

Remarks: In cluster with SPSR 25:422x844. Transform VIIIa.

174

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Order 27, reduced side 956

144178

25434

380 110

212125

239

128 114

22325252 ~~ 44

59~ 120

17472

115

324 343289

JDSII

Code: (380,254,144,178)(110,34)(212)(239,125)(114,11)(252, 128) (223)(59,174,120)(52,32,44)(20,12)(115)(72)(343)(324)(289)

Elements: 11, 12,20,32,34,44,52,59,72, 110, 114, 115, 120, 125, 128, 144, 174,178,212,223,239,252,254,289,324,343,380

Remarks: In cluster with C150. Transform I.

175

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Order 27, reduced side 960

492 468

127178 163

240 252 51 76

56 I40

I 123146 83

48 116

228 216 206168 142

Code: (492,468) (178, 127, 163) (240, 252) (51, 76) (40, 123) (146, 56, 27) (2,31,83)(29)(116)(228,12)(216,48)(206)(168,26)(142)

Elements: 2, 12,26,27,29,31,40,48,51,56,76,83, 116, 123, 127, 142, 146, 163, 168, 178,206,216,228,240,252,468,492

Remarks: in cluster with 208, A422. Transforms I, III, V.

176

CJB

Page 200: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 960

464496

87 99148 130

124

196 176 61 ~0154

72 52 6094 161149

50 44

268 288

199 205

CJB

Code: (496,464)(148,87,99,130)(196,124,176)(61, 26)(14,54, 31) (40)(161)(60,149)(94)(72,52)(288)(268)(50,44)(205)(199)

Elements: 14,26,31,40,44,50,52,54,60,61,72,87,94,99,124, 130,148,149,161,176,196,199,205,268,288,464,496

Remarks: In cluster with 153, A370. Transforms I, IlIa, V.

177

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Order 27, reduced side 960

436

524

196240

131

212181

44

8150

15246 38

68~

7969 54

66 55 284224

155 176121

Code: (524,436) (196, 240) (212,131,181) (152, 44) (81, 50) (284) (46, 38,68,79) (224,69)(8,30)(54)(32,66)(55,176)(155)(121)

Elements: 8,30,32,38,44,46,50,54,55,66,68,69,79,81,121,131, 152,155,176,181,196,212,224,240,284,436,524

Remarks: In cluster with 179, A414, A415. Transforms I, I, CJB-24, CJB-25, CJB-26, CJB­27.

178

CJB

Page 202: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 960

436

524

196240

139173

21244

73 66I 152

54Ur--

8761 68

38 46

50 63 284224

163 176113

Code: (524,436)(196,240)(212, 139,173)(152,44)(73,66)(284)(32,54,87)(68,30)(224,61)(8,46)(38)(63,176)(163,50)(113)

Elements: 8,30,32,38,44,46,50,54,61,63,66,68,73,87,113,139, 152,163, 173, 176,196,212,224,240,284,436,524

Remarks: In cluster with 178, A414, A415. Transforms I, I, CJB-24, CJB-25, CJB-26, CJB­27.

179

CJB

Page 203: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 964

429

535

188241

76132

~140

18753

55 M 13581 73 77

294242

216 212

Code: (535,429) (188, 241) (187, 132, 76, 140) (33,43) (135, 53) (23, 10) (32,21) (55,81,19)(294)(11,73,77)(62)(242)(216)(212)

CJB

Elements: 10,11,19,21,23,32,33;43,53,55,62,73,76,77,81,132, 135,140,187,188,212,216,241,242,294,429,535

Remarks: In cluster with A369. Transform I.

180

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Order 27, reduced side 965

277

355 333

78 75 56 68

hJ7 -I-J...=..:-

70 73 55 80109

39

257 321

148 239

52

353301 311

AJWD

Code: (355,277,333) (78, 75, 56, 68) (19,37) (12, 321) (80) (3, 73, 18) (257, 109, 70)(55)(39,239)(148)(353,52)(10,311)(301)

Elements: 3,10,12,18,19,37,39,52,55,56,68,70,73,75,78,80, 109,148,239,257,277,301,311,321,333,353,355

181

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Order 27, reduced side 966

261

354 351

90

171

64111 I---

190 164 47

266

158188

6539 99

151 54 50

58 62

349 346

271

AJWD

Cod~ (354,351,261)(90,171) (266,111,64)(190,164)(47,17) (188) (158)(65,99)(151,39)(54,50)(16,349)(346)(4,62)(58)(271)

Elements: 4, 16, 17,39,47,50,54,58; 62, 64, 65, 90, 99, 111, 151, 158, 164, 171, 188, 190,261,266,271,346,349,351,354

Remarks: In cluster with C163, C165. Transforms II, lIe, CJB-32.

182

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Order 27, reduced side 970

274336

360

71

,--

70 1 41 1 56 203

I68

99~

312288 60

211159

150 12888

322282

172 194

JDSII

Code: (336,360,274) (71, 203) (312, 24) (15,56) (288, 70, 41) (29, 68) (99) (60,8) (211)(159)(322,150,128)(88,282)(22,194)(172)

Elements: 8,15,22,24,29,41,56,60,68,70,71,88,99,128,150,159, 172,194,203,211,274,282,288,312,322,336,360

183

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Order 27, reduced side 970

274336 360

73

,.----

o l39J 60 2017 I r'----

68

312 101288 56

213157

150 12888

322282

172 194

AJWD

Code: (336,360,274) (73, 201) (312, 24) (13, 60) (288, 70, 39) (31, 8) (68) (101) (56,213)(157)(322,150,128)(88,282)(22,194)(172)

Elements: 8,13,22,24,31,.39,56,60,68,70,73,88,101,128,150,157, 172,194,201,213,274,282,288,312,322,336,360

184

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Order 27, reduced side 970

446

524

194252

129 149

24663

109 131

144 68

54162~272

200 199146 153

Code: (524,446) (252,194) (246,129,149) (63,131) (109, 20) (144, 25) (272,5) (68) (8,62,39) (200, 54) (199) (23,16) (7, 153) (146)

Elements: 5, 7, 8, 16,20,23,25,39,54,62,63,68, 109, 129, 131, 144, 146, 149, 153, 194,199,200,246,252,272,446,524

Remarks: In cluster with 169, A400, A427. Transforms I, Ib, II, III, IV, V.

185

CJB

Page 209: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 972

436

536

185251

128

211 19770

69 I 115

13792 117 13450 47 68

225 239189 168 151

AJWD

Code: (536,436) (185, 251) (211, 197, 128) (70, 115) (69, 32, 27) (117, 134) (5, 92) (37)(14,239,50) (47, 68) (225) (189) (168,17) (151)

Elements: 5,14,17,27,32,37,47,50,68,69,70,92,115,117,128, 134,151,168,185,189,197,211,225,239,251,436,536

186

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Order 27, reduced side 975

113159

268 67 146

435 88184

75119

162

43 120 13976

195 200245 1n 158

50

345 335295

JOSH

Code: (435,268,113,159)(67,46)(21,184)(88)(162,119, 75)(120,139) (43,76)(5,200)(245,195)(177,19)(158)(50,345)(335)(295)

Elements: 5,19,21,43,46,50,67,75,76,88,113,119,120,139,158, 159,162,177,184,195,200,245,268,295,335,345,435

Remarks: In cluster with 193, C145. Transforms I, la, IV.

187

Page 211: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 975

435

540

190245

168 140

232 55

135

37 65 8064 67

69 ~51-46 10245 - 295

203 215

134 128

CJB

Code: (540,435)(245, 190)(232, 168, 140)(55, 135)(65,295,80)(64,67, 37)(102)(215)(203,69,24)(21,46)(45)(20,26)(134)(128)

Elements: 20,21,24,26,37,45,46,55,64,65,67,69,80,102,128, 134,135,140,168,190,203,215,232,245,295,435,540

Remarks: In cluster with A387. Transfonn II.

188

Page 212: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 975

435

540

190245

164 140

236 55

13541 65 80

*57 45 58 106295

199 215142 124

Code: (540,435) (245,190) (236,164,140) (55,135) (65, 295, 80) (52, 71, 41) (106) (20,32)(215)(13,58)(199,57)(45)(142,18)(124)

Elements: 13,18,20,32,41,45,52,55,57,58,65,71,80,106,124, 135,140,142,164,190,199,215,236,245,295,435,540

Remarks: In cluster with A388. Transform II.

189

CJB

Page 213: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 976

471505

34

164138 130

273271

146 136 150

41

71 58106 118

200232

191129 '--

81 5687

AJWD

Code: (505,471) (34, 164,273) (271, 138, 130) (8, 136, 150) (146) (41, 232) (18, 118)(200,71) (191) (58,106) (129) (31,87) (81, 25) (56)

Elements: 8,18,25,31,34,41,56,58,71,81,87,106,118,129,130, 136,138,146,150,164,191,200,232,271,273,471,505

190

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Order 27, reduced side 979

298342 339

41

105 96141

257

3678 87

380

177135 98

189

91

70 95

319

235 260

165

AJWD

Code: (342,298,339) (257, 41) (380) (141, 105.96) (9, 87) (36, 78) (177) (135,30) (98,189)(7,91)(319)(25,95,260)(235,70)(165)

Elements: 7,9,25,30,36,41,70,78,87,91,95,96,98,105,135,141, 165, 177, 189,235,257,260,298,319,339,342,380

191

Page 215: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 980

254294

432

40

160214

174

~118

71 88 97222 181

203

104 85 79106

326 345309

AJWD

Code: (432,294,254) (40,214) (160,174) (222,118,47,23,22) (1,181) (24) (88,97,203)(71)(104,85)(79,9)(106)(345)(326)(309)

Elements: 1,9,22,23,24,40,47,71,79,85,88,97,104,106,118, 160,174,181,203,214,222,254,294,309,326,345,432

192

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Order 27, reduced side 980

113159

268 67 146

440 88184

75119

162

43 120 13976

195 205245 177 158

50

345 340295

JDSII

Code: (440,268,113,159) (67, 46)(21,184)(88)(162, 119,75) (120, 139) (43,76)(205)(245,195)(177,19)(158)(50,345,5)(340)(295)

Elements: 5, 19,21,43,46,50,67,75,76,88, 113, 119, 120, 139, 158, 159, 162, 177, 184,195,205,245,268,295,340,345,440

Remarks: In cluster with 187, C145. Transforms I, Ib, IV.

193

Page 217: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 982

,431

551

137 130164

44

1191---'- 116 123

200 188 59

~O~ 69177

33

298

231 243210 ,

Code: (551,431) (164,137,130) (200,119,188,44) (7,123) (28,116) (177,30,1) (29)(59)(50,69)(298)(31,19)(243,33)(231)(210)

Elements: 1,7,19,28,29,30,31,33,44,50,59,69,116,119,123,130, 137,164,177,188,200,210,231,243,298,431,551

Remarks: In cluster with 155, 167, A277, A346.

194

CJB

Page 218: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 984

433

551

187246

108 115 114

214 59

~ 83r- 128

101 94 -34

305

219 224 236

CJB

Code: (551,433) (187, 246) (214,108, 115, 114) (128,59) (101, 7) (1, 30, 83) (94, 29)(305)(31,28)(3,25)(34)(236)(5,224)(219)

Elements: 1,3,5,7,25,28,29,30,31,34,59,83,94,101,108,114, 115,128,187,214,219,224,236,246,305,433,551

Remarks: In cluster with A365. Transform I.

195

Page 219: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 985

459

526

109 131

93 219

87 65223 210 36 88

57~~

60 48 137

56

307

236 249193

AJWD

Code: (526,459) (109,131,219) (223, 210, 93) (87, 22) (65, 88) (57,36) (18,33,137) (3,15)(60)(307)(48)(13,249,56)(236)(193)

Elements: 3,13,15,18,22,33,36,48,56,57,60,65,87,88,93,109, 131,137,193,210,219,223,236,249,307,459,526

196

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Order 27, reduced side 985

435

550

115

320

182238 245

56 I 58 68

44 9050

81 ~r-- 7335 39

230197 190

116 112140

AJWD

Code: (550,435)(115,320)(238, 182,245) (56, 58,68) (90, 230) (]97, 81, 16)(14,44)(5,190,50)(73)(30)(35,39)(140)(1]6)(112)

Elements: 5, 14, 16, 30, 35, 39, 44, 50, 56, 58, 68, 73, 81,90, 112, 115, 116, 140, 182, 190,197,230,238,245,320,435,550

197

Page 221: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 987

459

528

199

132 260164

232

48 53100 --'=1'-- 98 -

38 50 45

156

105

268227 231

122 139

AJWD

Code: (528,459)(199,260)(232, 132, 164)(48,98,53)(100,32)(156,26,14)(12,50)(45,8) (38) (268) (231) (227,105) (17,139) (122)

Elements: 8,12,14,17,26,32,38,45,48,50,53,98,100,105,122, 132,139,156,164,199,227,231,232,260,268,459,528

198

Page 222: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 988

290 314384

94

172

338248 230 81 91

r--71

101

97 158 143151 103

54 43

336

205 216 231

JDSn

Code: (384,290,314) (94, 172,24) (338) (248, 230) (81, 91) (71, 10) (101) (158, 143)(151,97)(2,336)(103)(54,43)(15,231)(216)(205)

Elements: 2,10,15,24,43,54,71,81,91,94,97,101,103, ]43,151,158,172,205,216,230,231,248,290,314,336,338,384

199

Page 223: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 990

293 274

423

255

312

161

262 76

603'!J 63

179101

41 I 103

188

99 116

305 282

206 197

AJWD

Code: (423,293,274)(19,255)(312)(262,161) (76,179)(101,60)(34, 63,188,103)(5,29)(41,24)(116)(305,99)(282)(206,9)(197)

Elements: 5,9,19,24,29,34,41,60,63,76,99,101,103,116,161, 179,188,197,206,255,262,274,282,293,305,312,423

Remarks: In cluster with C179. Transform I.

200

Page 224: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 993

432

561

79 108'--~

50245

137143

225 243

39 66

104

105

53 ~~ 49 40 316

207 189136 145

Code: (561,432)(79,108,245)(50,29)(137)(225,243, 143)(66,316) (104,39)(105)(207,18)(189,53,19)(15,49,40)(34)(145)(136)

Elements: 15,18,19,29,34,39,40,49,50,53,66,79,104,105,108, 136,137,143,145,189,207,225,243,245,316,432,561

Remarks: In cluster with SPSR 25:432x864. Transform VIlI.

201

cm

Page 225: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1000

253308

439

93

64 160

131 113 r-- ~ 6749

144147

236205 268

79 l=-8958

51

356325 319

JDSII

Code: (439,308,253) (93,160) (131,113,64) (26, 67) (49, 41) (268) (18, 144) (236,147,205)(79,65)(89,58)(14,51)(356)(325)(319)

Elements: 14,18,26,41,49,51,58,64,65,67,79,89,93,113,131, 144,147,160,205,236,253,268,308,319,325,356,439

202

Page 226: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1004

336 304364

131118 71 173

90190

4267 231

141 16062

/------r 215195 ~

9788

409

283 312

JDSII

Code: (336,364,304)(131,173)(190, 118,28)(90,231,71)(160,42)(67,141)(215)(5,62)(195)(50,12)(38,409,97)(88)(312)(283)

Elements: 5,12,28,38,42,50,62,67,71,88,90,97,118,131,141, 160,173,190,195,215,231,283,304,312,336,364,409

203

Page 227: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1004

261

385 358

97164

120134 53

111251 58

115 335

117 57

12972

100

173152

368311

166 159

AJWD

Code: (385,261,358)(164,97)(120,335)(251, 134)(111,53)(58, 115) (117,17)(129,57)(72,100)(368)(173,28)(152,311)(166,7)(159)

Elements: 7,17,28,53,57,58,72,97,100,111,115,117,120,129, 134,152,159,164,166,173,251,261,311,335,358,368,385

204

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Order 27, reduced side 1005

275337

393

107168 45

127 15277 91 292

266

~139

88220

125167

147

52 42

346

251199 209

JDSn

Code: (393,275,337) (168, 107) (45,292) (152) (266, 127) (77, 91) (139, 51, 14) (37,220)(88)(125,167)(346,147)(52,251,42)(209)(199)

Elements: 14,37,42,45,51,52,77,88,91,107,125,127,139,147, 152,167,168,199,209,220,251,266,275,292,337,346,393

205

Page 229: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1008

280324

404

124156

111

213

110221

18927538

80 f--L...-

68 17554 ~'--

52 69

383329

296

AJWD

Code: (404,280,324) (124,156) (111, 213) (275, 221,32) (1,110) (189) (38,175) (80,30)(68)(54,383,35,18)(17,69)(52)(329)(296)

Elements: 1,17,18,30,32,35,38,52,54,68,69,80,110,111,124, 156,175,189,213,221,275,280,296,324,329,383,404

206

Page 230: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1008

272332

404

60

78 212

162 152

84147

251 57

165189 199

204 34

100 131151 97

51

325

253 228202

JDSH

Code: (404,332,272)(60,212)(78,162,152)(251,147,6)(84)(199, 165) (57, 189)(204)(34,131)(151,100)(325,97)(51,253)(228)(202)

Elements: 6,34,51,57,60,78,84,97,100,131,147,151,152,162, 165,189,199,202,204,212,228,251,253,272,325,332,404

207

Page 231: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1008

516 492

142 168206 240 252

116 48

83 146123 I 56

40 I76 51

276 264

163 178127

AJWD

Code: (516,492)(240,252) (206,142,168)(116,26)(146,48) (123,83) (276,12)(264)(31,29,56)(2,27)(40,76)(51,178)(163)(127)

Elements: 2,12,26,27,29,31,40,48,51,56,76,83,116,123,127, 142, 146, 163, 168, 178,206,240,252,264,276,492,516

Remarks: In cluster with 176, A422. Transforms I, III, V.

208

Page 232: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1020 JDSII

298 320

402

104172 107

43 23564

~.L...--

22553 69

281 157

67 55 60

56 175115

393337

290

Code: (402,298,320) (104,172,22) (107, 235) (281, 225) (43, 64) (157, 37,21) (16, 69)(53)(60,175)(67,55)(115)(56,393)(337)(290)

Elements: 16,21,22,37,43,53,55,56,60,64,67,69,104,107,115, 157,172,175,225,235,281,290,298,320,337,393,402

209

Page 233: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1021

284 308

429

260

332155

27476 88 9636

I....--119

~~ 631

83 55

158 164

381318

160 162

JDSII

Code: (429,284,308)(260,24)(332)(274,155)(76, 88,96)(119, 36)(83,29)(17,63,8) (55, 381) (46) (164) (318,158) (2,162) (160)

Elements: 2,8,17,24,29,36,46,55,63,76,83,88,96,119,155,158, 160,162,164,260,274,284,308,318,332,381,429

210

Page 234: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1022

461

561

182

279

166 167228 85 97

5783 ~~109 106 94

52 43

78 55282

233186 188

133

AJWD

Code: (561,461) (182, 279) (228,166,167) (85, 97) (57,109) (83, 63, 106) (94,282) (5,52)(233)(20,43)(25,78)(55,188)(186)(133)

Elements: 5,20,25,43,52,55,57,63, 78,83,85,94,97, 106, 109, 133, ]66,167, 182, 186,188,228,233,279,282,461,561

211

Page 235: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1023

433

590

224 209

117 90148 41 56

235I 153

55186 113 160 97

124

37

198 230 206 228161

JDSII

Code: (590,433)(224,209)(235,148, 117,90)(56,153)(23,160,41) (113) (97)(31, 86)(124,55)(230,24)(22,228)(206)(198,37)(161)

Elements: 22,23,24,31,37,41,55,56,86,90,97, 113, 117, 124, 148, 153, 160, 161, 198,206,209,224,228,230,235,433,590

212

Page 236: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1028

267

351410

84

183

148

165 58287

125 t---245 67

139

63132

16252 99

193154

390

291

180 167

AJWD

Code: (410,267,351) (183,84) (148, 287) (245, 165) (125, 58) (9, 139) (67) (132, 162,63)(99,390)(193,52)(154,30)(291)(180,13)(167)

Elements: 9,13,30,52,58,63,67,84,99,125,132,139,148,154, 162,165,167,180,183,193,245,267,287,291,351,390,410

213

Page 237: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1032

272

356404

89

183

9451

10150

252 351

151 278

59

164 ~180 ~64

299 325

196 212

JDSII

Code: (404,272,356) (183,89) (5,351) (94) (252,101,51) (50,278) (151) (180,164,59)(26,325)(41,18)(23,299)(64)(16,212)(196)

Elements: 5, 16, 18,23,26,41,50,51,59,64,89,94, 101,151, 164,180, 183, 196,212,252,272,278,299,325,351,356,404

214

Page 238: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1038

331 313

394

63

286 295170

287

53117

64 94100

175 201111 79

115 1--.-

40 51

357

246 229206

AJWD

Code: (394,331,313) (18, 295) (63, 286) (287,170) (117,53) (94, 201) (64,175,100)(357,111)(15,79)(115)(51,28)(246,40)(229)(206)

Elements: 15,18,28,40,51,53,63,64,79,94,100,111,115,117, 170, 175,201,206,229,246,286,287,295,313,331,357,394

215

Page 239: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1040

500540

120 12796

253236 208 6197 73

112 r-r-r-r-1

5352

126 ~71

264 292 287

197

JDSU

Code: (540,500) (120, 127,253) (236, 208, 96) (16,97,7) (61, 73) (112) (29,32) (20, 53)(126)(52)(28,292)(19,287)(71)(264)(197)

Elements: 7, 16, 19,20,28,29,32,52,53,61, 71, 73, 96, 97, 112, 120, 126, 127, 197,208,236,253,264,287,292,500,540

216

Page 240: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1050

302

381 367

79 89134

143

70 ;-164

84 r;;109

296 324

132 111

187

7655

373 359318

AJWD

Code: (381,302,367)(79,134,89)(43,324)(296, 164)(70, 19)(62)(109,25)(84,11)(73)(132,187,111)(76,35)(373,55)(359)(318)

Elements: 11,19,25,35,43,55,62,70,73,76,79,84,89,109,111, 132,134,164,187,296,302,318,324,359,367,373,381

217

Page 241: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1050

269

367414

98171

73

197 195141

273 244

-Tl 53132 122

165114

42

363 366321

JDSII

Code: (414,367,269) (98,171) (197,195,73) (273,141) (244) (132, 9) (20,53,122) (165,23,18)(5,33)(28)(114)(366)(363,42)(321)

Elements: 5,9,18,20,23,28,33,42,53,73,98,114,122,132,141, 165,171,195,197,244,269,273,321,363,366,367,414

218

Page 242: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1060

491

569

226139

265164

266I 43 39

46 1 71 IT:;143 ~ 153

124 7041

304

225184 157

190

AJWD

Code: (569,491)(226,265)(266,164,139)(25, 71,43)(34,153,39)(143,46)(28,49)(304)(124,21)(70)(225,41)(33,190)(184)(157)

Elements: 21,25,28,33,34,39,41,43,46,49,70,71,124, ]39, 143,153,157,164,184,190,225,226,265,266,304,491,569

219

Page 243: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1064

266

416382

150116

151185

159

256 313

128167 200

7072

18682

130

148369

297

206 192

JDSII

Code: (416,266,382)(150,116)(185,313)(256,159, 151)(8,200, 128) (167)(186,70)(72,369)(82,130,25)(297)(34,48)(206,14)(192)

Elements: 8,14,25,34,48,70,72,82,116,128,130,150,151,159, 167,185,186,192,200,206,256,266,297,313,369,382,416

220

Page 244: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1064

500564

198134

173 302

257

8256 l- 106

84 I---l-144

92 38 4079 6898

47

243 262192 222

145

AJWD

Code: (564,500)(198,302)(257,173,134)(82, 106, 144)(84,56,33)(23, 10)(92)(79)(40,262)(68,38)(243,98)(222)(47,192)(145)

Elements: 10,23,33,38,40,47,56,68,79,82,84,92,98,106,134,144, 145,173,192,198,222,243,257,262,302,500,564

221

Page 245: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1075

278

379418

140 138

37

71 148 50170 172

f.------l-. 125165 75

94274

109 192 176 174150

383350342

AJWD

Code: (418,278,379) (140,138) (37,170,172) (50,125) (274,165,71,48) (23,75) (94)(192,8) (176, 2) (174) (109,150) (383) (350) (342)

Elements: 2,8,23,37,48,50, 71, 75, 94, 109, 125,138, 140, 150, 165, 170, 172, 174, 176,192,274,278,342,350,379,383,418

222

Page 246: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1085

325 328

432

107~

102165 64

21562

10444

241298 183

139

224

5794

412355

318

JDSII

Code: (432,325,328)(107,165,39, 14)(11,102,215)(25)(64)(62,2) (104)(298,241)(183,44)(139,9)(224)(57,412,94)(355)(318)

Elements: 2,9,11,14,25,39,44,57,62,64,94,102,104,107,139, 165,183,215,224,241,298,318,325,328,355,412,432

223

Page 247: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1091

310 335

446

92

cr;- 193

53~141 360

88 84262 156

121192

120

383 396

312

AJWD

Code: (446,310,335)(92, 193,25)(360)(34,58)(10,24)(262, 141,53) (39,43)(88,4)(84,156)(121,192)(120,36)(396)(383)(312)

Elements: 4, 10,24,25,34,36,39,43,53,58,84,88,92, 120,121,141, 156,192, 193,262,310,312,335,360,383,396,446

224

Page 248: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1111

324 340

447

123185

119149 35657

128302

93 ~71

105

59

60 106140

88

415362

334

AJWD

Code: (447,324,340)(123,185,16)(356)(302,149, 119)(57, 128)(105,71)(93,56)(140,59)(37,19)(18,106)(60,88)(415)(362)(334)

Elements: 16, 18, 19, 37, 56, 57, 59, 60, 71, 88, 93, 105, 106, 119, 123, 128, 140, 149, 185,302,324,334,340,356,362,415,447

225

Page 249: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1123

302367

454

88 65

~149

84

151

140

165204 348128

278

113 60180

144

391 408

324

JDSII

Code: (454, 302, 367) (88, 149,65) (84, 348) (38, 50) (26, 12) (51, 11) (40, 204) (278,165,37)(128)(113,180)(144,60)(408)(391)(324)

Elements: 11,12,26,37,38,40,50,51,60,65,84,88,113,128,144, 149,165,180,204,278,302,324,348,367,391,408,454

226

Page 250: Simple perfect squared squares of order 27 · Simple perfect squared squares of order 27 Citation for published version (APA): Bouwkamp, C. J. (1998). Simple perfect squared squares

Order 27, reduced side 1143

308343

492

273

88 378148

256 60 ~'-55 106

77~140

108

130100 96

395 422

326

JDSn

Code: (492,308,343)(273,35)(378)(256,148,88)(60,28)(27, 106,140) (55)(108,77,23)(54,24)(130)(31,100)(96,422)(395)(326)

Elements: 23,24,27,28,31,35,54,55,60,77,88,96,100,106,108, 130,140,148,256,273,308,326,343,378,395,422,492

227