Affine Magic Squares–Independently Discovered

Benjamin Franklin's Magic Squares

Benjamin Franklin made a few magic squares with interesting and unusual properties; at least four of them have survived. Two are well-known, and are explained in Chapter III of Andrews. Another was published by Pasles. Ahmed has studied the methods by which Franklin might have constructed these squares; she gives some more examples of squares constructed by her methods.

A Square of order 8

Franklin's first square is:

52	61	4	13	20	29	36	45
14	3	62	51	46	35	30	19
53	60	5	12	21	28	37	44
11	6	59	54	43	38	27	22
55	58	7	10	23	26	39	42
9	8	57	56	41	40	25	24
50	63	2	15	18	31	34	47
16	1	64	49	48	33	32	17

It has the following magical properties:

Each row and column adds up to the magic sum, which is 260.
Every half-row and half-column adds up to half the magic sum; for instance, 52+61+4+13=130, or 46+35+30+19=130, or 4+62+5+59=130, or 23+41+18+48=130.
Each bent row or bent column adds to the magic sum. Two bent rows are shown in Figure 1; others can be formed by moving translating these up or down. Likewise, two bent columns are shown in Figure 2, and these can be translated horizontally. If the translated figure would pass through an edge of the square, it should wrap around to the opposite edge; for instance, 36+19+53+6+58+9+47+32=260 is a bent column.
Every 2-by-2 subsquare adds up to half the magic sum. This property also wraps around at the edges of the square, as with 19+44+14+53=130 or 1+64+61+4=130, or even wrapping around both horizontally and vertically: 52+45+16+17=130.

Figure 1. Bent rows

Figure 2. Bent columns

This square is affine. Here are the affine functions which define it:

A₁ = f₃+f₄+f₅+1

A₂ = f₃+f₅+1

A₃ = f₃+f₆+0

A₄ = f₁+f₂+f₃+f₆+0

A₅ = f₂+f₃+f₆+1

A₆ = f₁+f₂+f₆+1

It is easy enough to check that the square is uniform on the subspace <e₂, e₃>, which is a half column, and consequently the sum is 130 on every parallel flat, which is to say, on every half column. Likewise for the half rows. But what about these bent rows and bent columns? They can be investigated using the applet that displays flats. It turns out that half of the bent rows and columns actually are 3-flats, but the others are not. To understand the bent rows, you can use the two linear subspaces

α = <e₂+ e₅, e₄ + e₅ + e₆>

β = <e₁+e₂+e₄, e₄ + e₅ + e₆>.

The affine square is uniform on both of these. Every flat that is parallel to α or to β is part of a bent row; and every bent row consists of two such flats. So the magical property of bent rows follows from the affine structure of the square. A similar analysis works for the bent columns.

As for the 2×2 blocks, they are all 2-flats. They belong to four families of parallel flats, not all of which are contiguous 2×2 blocks. Franklin hinted at "other" properties of this square, and it occurs to me that he may have known about these other 2-flats.

Is there any chance that Franklin constructed this square, and others like it, by defining an affine map with the requisite properties? Certainly he could not have used the concepts and notation of linear algebra, which did not exist in his day. But we do not know what technique he did use. One possible technique is sketched in Chapter III of Andrews. This is, to start with the numbers in their natural order, which corresponds to the identity transform as an affine map, and apply various transformations which preserve the affine property. In Andrews, figures 194 - 197, you can see a nontrivial affine map being constructed in this way.

A Lesser-known square of order 8

This square was relegated to a footnote in Franklin's collected works. It is resurrected by Pasles and further treated by Ahmed.

17	47	30	36	21	43	26	40
32	34	19	45	28	38	23	41
33	31	46	20	37	27	42	24
48	18	35	29	44	22	39	25
49	15	62	4	53	11	58	8
64	2	51	13	60	6	55	9
1	63	14	52	5	59	10	56
16	50	3	61	12	54	7	57

It, too, is affine. defined by these functions:

A₁ = f₁+f₂+f₆+0

A₂ = f₂+f₆+1

A₃ = f₃+f₅+f₆+0

A₄ = f₃+f₄+f₅+f₆+0

A₅ = f₃+f₆+0

A₆ = f₃+f₅+0

Squares of order 16

One square of order 16, constructed by Franklin, has been reprinted in several places, including Andrews:

200	217	232	249	8	25	40	57	72	89	104	121	136	153	168	185
58	39	26	7	250	231	218	199	186	167	154	135	122	103	90	71
198	219	230	251	6	27	38	59	70	91	102	123	134	155	166	187
60	37	28	5	252	229	220	197	188	165	156	133	124	101	92	69
201	216	233	248	9	24	41	56	73	88	105	120	137	152	169	184
55	42	23	10	247	234	215	202	183	170	151	138	119	106	87	74
203	214	235	246	11	22	43	54	75	86	107	118	139	150	171	182
53	44	21	12	245	236	213	204	181	172	149	140	117	108	85	76
205	212	237	244	13	20	45	52	77	84	109	116	141	148	173	180
51	46	19	14	243	238	211	206	179	174	147	142	115	110	83	78
207	210	239	242	15	18	47	50	79	82	111	114	143	146	175	178
49	48	17	16	241	240	209	208	177	176	145	144	113	112	81	80
196	221	228	253	4	29	36	61	68	93	100	125	132	157	164	189
62	35	30	3	254	227	222	195	190	163	158	131	126	99	94	67
194	223	226	255	2	31	34	63	66	95	98	127	130	159	162	191
64	33	32	1	256	225	224	193	192	161	160	129	128	97	96	65

Like his squares of order 8, this is magic for rows, columns, bent rows, and bent columns, the magical sum being 2056. It is also magical for all contiguous 4×4 blocks; this is a corollary of the fact that every contiguous 2×2 block adds up to 514. Franklin affirmed this "to be the most magically magical of any magic square ever made by any magician." But only up to that time, which was about 1752. This square is affine, and here are its defining functions.

A₁ = f₄+f₅+f₆+1

A₂ = f₄+f₆+1

A₃ = f₄+f₇+0

A₄ = f₄+f₈+0

A₅ = f₁+f₂+f₄+f₈+0

A₆ = f₂+f₄+f₈+1

A₇ = f₁+f₂+f₃+f₄+f₈+1

A₈ = f₁+f₂+f₈+1

In a 1765 letter to the electrical investigator John Canton, Franklin included another square of order 16, which was first published by Pasles and can best be seen at his web site. It is:

16	255	2	241	14	253	4	243	12	251	6	245	10	249	8	247
1	242	15	256	3	244	13	254	5	246	11	252	7	248	9	250
240	31	226	17	238	29	228	19	236	27	230	21	234	25	232	23
225	18	239	32	227	20	237	30	229	22	235	28	231	24	233	26
223	48	209	34	221	46	211	36	219	44	213	38	217	42	215	40
210	33	224	47	212	35	222	45	214	37	220	43	216	39	218	41
63	208	49	194	61	206	51	196	59	204	53	198	57	202	55	200
50	193	64	207	52	195	62	205	54	197	60	203	56	199	58	201
80	191	66	177	78	189	68	179	76	187	70	181	74	185	72	183
65	178	79	192	67	180	77	190	69	182	75	188	71	184	73	186
176	95	162	81	174	93	164	83	172	91	166	85	170	89	168	87
161	82	175	96	163	84	173	94	165	86	171	92	167	88	169	90
159	112	145	98	157	110	147	100	155	108	149	102	153	106	151	104
146	97	160	111	148	99	158	109	150	101	156	107	152	103	154	105
127	144	113	130	125	142	115	132	123	140	117	134	121	138	119	136
114	129	128	143	116	131	126	141	118	133	124	139	120	135	122	137

Of course, it too is affine; the defining functions are:

A₁ = f₂+f₃+f₈+0

A₂ = f₁+f₂+f₃+f₈+0

A₃ = f₃+f₈+0

A₄ = f₂+f₈+0

A₅ = f₄+f₇+1

A₆ = f₄+f₅+f₇+1

A₇ = f₄+f₆+f₇+1

A₈ = f₂+f₄+f₈+1

And this one is even more magical than the previous square; in particular, it is pandiagonal.

How many magic properties has an affine square?

We can see that the magic properties of Franklin's squares can be derived from their being (a) affine, and (b) uniform on certain linear subspaces of various dimensions. On how many subspaces is an affine square uniform? Here is a "calculator" which gives the answer: the number of uniform subspaces of dimension m, and the number of m-flats parallel to those spaces, in a vector space F_p^d.

P:		This selector gives you the order of the prime field.
D:		This selector chooses the dimension of the vector space.
M:		This selector chooses the dimension m of the linear subspaces and flats being counted.

The number of cells is 000.

There are 000 uniform linear subspaces of dimension 0, and 000 uniform 0-flats.

The calculator starts up with p = 2, d = 4, and m = 2, which pertain to such squares as the Dürer square. Try p = 2, d = 8, and m = 4 to see how much magic there is in one of Franklin's squares. Franklin's "most magically magical" statement was in fact modest.

The theory behind this calculator is described in some notes, along with the theory of counting affine squares of any prime-power order.

Great Multimagic Squares

Benson and Jacoby constructed a trimagic square of order 32. This is the smallest trimagic square of prime-power order that I know of. I investigated it using Mathematica™. It is an affine square. The rows and columns are uniform of degree 3, but the diagonals are not. However, they are not very non-uniform. In degree two, there are two pairs of digit coordinates that are not linearly independent on the diagonal subspace: A₃A₁₀ and A₄A₉. On the main diagonal, most of the products of two digit coordinates take the value 1 in 1/4 of the cells; but A₃A₁₀ has the value 1 nowhere, and A₄A₉ has the value 1 at 1/2 of the cells. Note that 3 + 10 = 4 + 9; these two products together contribute as much to the sum of squares as if they were both uniform. Similarly, on the opposite diagonal, A₄A₉ contributes nothing whereas A₃A₁₀ contributes a double amount.

Christian Boyer has constructed a web site with much information on multimagic squares. He provides downloadable copies of several multimagic squares of impressively high degree. I have investigated them to see if they were affine. Here are some findings.

Trimagic, order 64 The first such square was made by General Eutrope Cazalas, in 1933. It is affine, and is uniform of degree 3 in the rows, columns, and diagonals.

Trimagic, order 128 The first such square was made by Gaston Tarry in 1905. It is not affine, but the discrepancy is less than it might be. My method of checking for the affine property was to find the affine square which would agree with Tarry's square at 15 points, namely those corresponding to {0 , e₁,..., e₁₄}. I compared the digit coordinates of this square with those of Tarry's. There were 14×16384 pairs of bits to check. The two squares differend in 3×16384 cases. Six of the digit coordinates agreed completely. As for the others, two were discrepant at 5120 places, four at 6144 places, and two at 7168 places. Most surely, Tarry's square has some hidden structure; it is not as simple as an affine square, but it is not very much more complicated.

Tetramagic, order 243 A square of this description was constructed by Pan Fengchu in 2004; this appears to be the smallest tetramagic square currently known. Alas, it is not affine. I compared it with the affine square determined by 11 points, and found discrepancies at somewhat more than 50% of the data.

Last modified on $Date: 2010-12-11 13:34:00 -0500 (Sat, 11 Dec 2010) $

Christopher J. Henrich

52	61	4	13	20	29	36	45
14	3	62	51	46	35	30	19
53	60	5	12	21	28	37	44
11	6	59	54	43	38	27	22
55	58	7	10	23	26	39	42
9	8	57	56	41	40	25	24
50	63	2	15	18	31	34	47
16	1	64	49	48	33	32	17

17	47	30	36	21	43	26	40
32	34	19	45	28	38	23	41
33	31	46	20	37	27	42	24
48	18	35	29	44	22	39	25
49	15	62	4	53	11	58	8
64	2	51	13	60	6	55	9
1	63	14	52	5	59	10	56
16	50	3	61	12	54	7	57

200	217	232	249	8	25	40	57	72	89	104	121	136	153	168	185
58	39	26	7	250	231	218	199	186	167	154	135	122	103	90	71
198	219	230	251	6	27	38	59	70	91	102	123	134	155	166	187
60	37	28	5	252	229	220	197	188	165	156	133	124	101	92	69
201	216	233	248	9	24	41	56	73	88	105	120	137	152	169	184
55	42	23	10	247	234	215	202	183	170	151	138	119	106	87	74
203	214	235	246	11	22	43	54	75	86	107	118	139	150	171	182
53	44	21	12	245	236	213	204	181	172	149	140	117	108	85	76
205	212	237	244	13	20	45	52	77	84	109	116	141	148	173	180
51	46	19	14	243	238	211	206	179	174	147	142	115	110	83	78
207	210	239	242	15	18	47	50	79	82	111	114	143	146	175	178
49	48	17	16	241	240	209	208	177	176	145	144	113	112	81	80
196	221	228	253	4	29	36	61	68	93	100	125	132	157	164	189
62	35	30	3	254	227	222	195	190	163	158	131	126	99	94	67
194	223	226	255	2	31	34	63	66	95	98	127	130	159	162	191
64	33	32	1	256	225	224	193	192	161	160	129	128	97	96	65

16	255	2	241	14	253	4	243	12	251	6	245	10	249	8	247
1	242	15	256	3	244	13	254	5	246	11	252	7	248	9	250
240	31	226	17	238	29	228	19	236	27	230	21	234	25	232	23
225	18	239	32	227	20	237	30	229	22	235	28	231	24	233	26
223	48	209	34	221	46	211	36	219	44	213	38	217	42	215	40
210	33	224	47	212	35	222	45	214	37	220	43	216	39	218	41
63	208	49	194	61	206	51	196	59	204	53	198	57	202	55	200
50	193	64	207	52	195	62	205	54	197	60	203	56	199	58	201
80	191	66	177	78	189	68	179	76	187	70	181	74	185	72	183
65	178	79	192	67	180	77	190	69	182	75	188	71	184	73	186
176	95	162	81	174	93	164	83	172	91	166	85	170	89	168	87
161	82	175	96	163	84	173	94	165	86	171	92	167	88	169	90
159	112	145	98	157	110	147	100	155	108	149	102	153	106	151	104
146	97	160	111	148	99	158	109	150	101	156	107	152	103	154	105
127	144	113	130	125	142	115	132	123	140	117	134	121	138	119	136
114	129	128	143	116	131	126	141	118	133	124	139	120	135	122	137

52	61	4	13	20	29	36	45
14	3	62	51	46	35	30	19
53	60	5	12	21	28	37	44
11	6	59	54	43	38	27	22
55	58	7	10	23	26	39	42
9	8	57	56	41	40	25	24
50	63	2	15	18	31	34	47
16	1	64	49	48	33	32	17

17	47	30	36	21	43	26	40
32	34	19	45	28	38	23	41
33	31	46	20	37	27	42	24
48	18	35	29	44	22	39	25
49	15	62	4	53	11	58	8
64	2	51	13	60	6	55	9
1	63	14	52	5	59	10	56
16	50	3	61	12	54	7	57

200	217	232	249	8	25	40	57	72	89	104	121	136	153	168	185
58	39	26	7	250	231	218	199	186	167	154	135	122	103	90	71
198	219	230	251	6	27	38	59	70	91	102	123	134	155	166	187
60	37	28	5	252	229	220	197	188	165	156	133	124	101	92	69
201	216	233	248	9	24	41	56	73	88	105	120	137	152	169	184
55	42	23	10	247	234	215	202	183	170	151	138	119	106	87	74
203	214	235	246	11	22	43	54	75	86	107	118	139	150	171	182
53	44	21	12	245	236	213	204	181	172	149	140	117	108	85	76
205	212	237	244	13	20	45	52	77	84	109	116	141	148	173	180
51	46	19	14	243	238	211	206	179	174	147	142	115	110	83	78
207	210	239	242	15	18	47	50	79	82	111	114	143	146	175	178
49	48	17	16	241	240	209	208	177	176	145	144	113	112	81	80
196	221	228	253	4	29	36	61	68	93	100	125	132	157	164	189
62	35	30	3	254	227	222	195	190	163	158	131	126	99	94	67
194	223	226	255	2	31	34	63	66	95	98	127	130	159	162	191
64	33	32	1	256	225	224	193	192	161	160	129	128	97	96	65

16	255	2	241	14	253	4	243	12	251	6	245	10	249	8	247
1	242	15	256	3	244	13	254	5	246	11	252	7	248	9	250
240	31	226	17	238	29	228	19	236	27	230	21	234	25	232	23
225	18	239	32	227	20	237	30	229	22	235	28	231	24	233	26
223	48	209	34	221	46	211	36	219	44	213	38	217	42	215	40
210	33	224	47	212	35	222	45	214	37	220	43	216	39	218	41
63	208	49	194	61	206	51	196	59	204	53	198	57	202	55	200
50	193	64	207	52	195	62	205	54	197	60	203	56	199	58	201
80	191	66	177	78	189	68	179	76	187	70	181	74	185	72	183
65	178	79	192	67	180	77	190	69	182	75	188	71	184	73	186
176	95	162	81	174	93	164	83	172	91	166	85	170	89	168	87
161	82	175	96	163	84	173	94	165	86	171	92	167	88	169	90
159	112	145	98	157	110	147	100	155	108	149	102	153	106	151	104
146	97	160	111	148	99	158	109	150	101	156	107	152	103	154	105
127	144	113	130	125	142	115	132	123	140	117	134	121	138	119	136
114	129	128	143	116	131	126	141	118	133	124	139	120	135	122	137

Affine Magic Squares Independently Discovered