Most-perfect Bent Diagonal Magic Squares

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 ½n 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