Most-perfect Bent Diagonal Magic Squares

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On March 22, 2007 I first heard about a new magic square discovery in The Netherlands

Three Dutch secondary school pupils have created the ‘most magical magic square in 5,000 years’,… [1]

A lively discussion among magic square friends ensued over the next week. The  result? While these students should be complimented on their accomplishment, their imaginative claim was slightly (?) exaggerated.

The square in question is a bent-diagonal (Franklin-type) order-12 square with rows and columns summing correctly. Also present are many other magic patterns found in Franklin type squares. The creators excitement seemed to be due to the fact that the main diagonals also summed correctly, making this a true magic square. This was a feature Ben Franklin did not accomplish in his published squares.

It is also symmetric across the central horizontal line.

I found this square with identical features to the HSA square at Donald Morris’s Franklin Squares site [2].

This was published in 2005 and included a complete method of construction!

The HSA square is identical to this square reflected across the leading diagonal !!!!

Well... after first swapping columns 5 and 7, then 6 and 8. (This pointed out by Jo Geuskens on May 27/07 and  Frans Lelieveld on May 30/07. Thanks fellows.)

 

Finally, I compared the HAS square to an order-12 Most-perfect magic square on my site. [3]

In the comparison, my square fails on

  • Bent diagonals (and many other Franklin patterns)

  • Horizontal (and vertical) lines of 4

  • Vertical symmetry (across the center horizontal line)

My square also is

  • Pandiagonal

  • Compact (2x2 squares sum to 4/12 of S)

  • And it is Most-perfect

The HSA (and the Morris) square fails on the most-perfect diagonal test!

Strangely, if my square is transposed so that the 1 is in the upper left corner, all horizontal bent diagonals become magic! 

Order-12  HSA Square

1

142

11

136

8

138

5

139

12

135

2

141

120

27

110

33

113

31

116

30

109

34

119

28

121

22

131

16

128

18

125

19

132

15

122

21

48

99

38

105

41

103

44

102

37

106

47

100

73

70

83

64

80

66

77

67

84

63

74

69

60

87

50

93

53

91

56

90

49

94

59

88

85

58

95

52

92

54

89

55

96

51

86

57

72

75

62

81

65

79

68

78

61

82

71

76

97

46

107

40

104

42

101

43

108

39

98

45

24

123

14

129

17

127

20

126

13

130

23

124

25

118

35

112

32

114

29

115

36

111

26

117

144

3

134

9

137

7

140

6

133

10

143

4

Order-12  from D. Morris page

1

120

121

48

85

72

73

60

97

24

25

144

142

27

22

99

58

75

70

87

46

123

118

3

11

110

131

38

95

62

83

50

107

14

35

134

136

33

16

105

52

81

64

93

40

129

112

9

8

113

128

41

92

65

80

53

104

17

32

137

138

31

18

103

54

79

66

91

42

127

114

7

5

116

125

44

89

68

77

56

101

20

29

140

139

30

19

102

55

78

67

90

43

126

115

6

12

109

132

37

96

61

84

49

108

13

36

133

135

34

15

106

51

82

63

94

39

130

111

10

2

119

122

47

86

71

74

59

98

23

26

143

141

28

21

100

57

76

69

88

45

124

117

4

Order-12 from my Most-perfect page

65

93

82

95

49

78

68

64

51

62

84

79

32

100

15

98

48

115

29

129

46

131

13

114

25

133

42

135

9

118

28

104

11

102

44

119

24

108

7

106

40

123

21

137

38

139

5

122

17

141

34

143

1

126

20

112

3

110

36

127

76

56

59

54

92

71

73

85

90

87

57

70

77

81

94

83

61

66

80

52

63

50

96

67

116

16

99

14

132

31

113

45

130

47

97

30

117

41

134

43

101

26

120

12

103

10

136

27

124

8

107

6

140

23

121

37

138

39

105

22

125

33

142

35

109

18

128

4

111

2

144

19

72

60

55

58

88

75

69

89

86

91

53

74


Announcement!

On April 2, 2007 I received an email from Donald Morris with an order 16 Most-perfect Bent diagonal magic square! [4]

Even more surprising was this order 12 square also included in the attachment. It also is a Most-perfect Bent diagonal magic square!

To the best of my knowledge, these are the first such squares published!
Don tells me he constructed this square in late 2005.

 

Order-12 From Morris email of April 2,2007

1

120

85

72

97

24

133

36

49

84

37

132

142

27

58

75

46

123

10

111

94

63

106

15

8

113

92

65

104

17

140

29

56

77

44

125

138

31

54

79

42

127

6

115

90

67

102

19

9

112

93

64

105

16

141

28

57

76

45

124

134

35

50

83

38

131

2

119

86

71

98

23

12

109

96

61

108

13

144

25

60

73

48

121

135

34

51

82

39

130

3

118

87

70

99

22

5

116

89

68

101

20

137

32

53

80

41

128

139

30

55

78

43

126

7

114

91

66

103

18

4

117

88

69

100

21

136

33

52

81

40

129

143

26

59

74

47

122

11

110

95

62

107

14

Order-16

On April 2, 2007, Donald Morris sent me this order-16 magic square that has almost all of the features (68) found in Franklin's unpublished order-16 on my Franklin page. [5]

And this one is Most-perfect!
Donald reports that he constructed this square in late 2005

 

The Morris Order-16 Most-perfect Bent diagonal magic square

256

225

48

49

80

81

160

129

16

17

224

193

192

161

112

113

15

18

223

194

191

162

111

114

255

226

47

50

79

82

159

130

243

238

35

62

67

94

147

142

3

30

211

206

179

174

99

126

4

29

212

205

180

173

100

125

244

237

36

61

68

93

148

141

245

236

37

60

69

92

149

140

5

28

213

204

181

172

101

124

6

27

214

203

182

171

102

123

246

235

38

59

70

91

150

139

250

231

42

55

74

87

154

135

10

23

218

199

186

167

106

119

9

24

217

200

185

168

105

120

249

232

41

56

73

88

153

136

241

240

33

64

65

96

145

144

1

32

209

208

177

176

97

128

2

31

210

207

178

175

98

127

242

239

34

63

66

95

146

143

254

227

46

51

78

83

158

131

14

19

222

195

190

163

110

115

13

20

221

196

189

164

109

116

253

228

45

52

77

84

157

132

252

229

44

53

76

85

156

133

12

21

220

197

188

165

108

117

11

22

219

198

187

166

107

118

251

230

43

54

75

86

155

134

247

234

39

58

71

90

151

138

7

26

215

202

183

170

103

122

8

25

216

201

184

169

104

121

248

233

40

57

72

89

152

137

 Order-8

Recently Daniel Schindel, Matthew Rempel And Peter Loly (Winnipeg, Canada) counted the basic Franklin type bent-diagonal squares of order-8. [6]
There are exactly 1,105,920 of them. Two-thirds of these squares are not magic because the main diagonals do not sum correctly. Exactly one-third (368,640) are pandiagonal magic.

BTW The Peter Loly's count has been independently corroborated by other sources in Canada and Argentina.
This figure (368,640) is in exact agreement with that reported by Dame Kathleen Ollerenshaw as being most-perfect. The bent-diagonal pandiagonal squares all have the 2z2 feature (compact), but fail on the diagonal feature (complete) so we can assume that there are no order-8 bent-diagonal most-perfect magic squares!

Review of requirements to be classed as most-perfect:

1.      Doubly-even pandiagonal normal magic squares (i.e. order 4, 8, 12, etc using integers from 1 to n2)

2.      Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n2 + 1) (compact)

3.      Any pair of integers distant ˝n along a diagonal sum to T (complete)

[1] Announcement of the HSA square http://www.eurogates.nl/?act=shownews&nid=1856
[2] Donald Morris's Franklin squares site http://www.bestfranklinsquares.com/
[3] My Most-perfect magic squares page
[4] Donald Morris's email address (with his permission) is donald.morris4@sbcglobal.net
[5] My Franklin magic squares page
[6] Proc. R. Soc. A (2006) 462, 2271–2279, doi:10.1098/rspa.2006.1684. Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS
Published online 28 February 2006. Obtainable by download from Peter Loly's home page

 

This page was originally posted April 2007
It was last updated October 22, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz